We here demolish the argument used by Godel to claim that Arithmetic is incomplete. We decided for splitting our paper on Godel to make it easier for us in the so horrible situation we face since end of 2001 because of Brazil, its government, majorly, but also because of Australian services for law and order. We really wish that AU were a sane place, with sane people, but things were clearly not that way for us already in 2001. We fear things may get even worse than they are. Currently, the AU police is denying having ever received any denouncements from us until this year (NSW police gave us a reference number and the name of a cop only this year, but denouncements are made with AFP, for instance, since 2004, first one alerted having been AFP VIC, and Glenn Woodard, from AFP QLD has got more than 3 pages of material proof with him. AFP has denied having received any denouncement from us recently, in 2011, when we had this great idea of first making sure that everything was in their computers by means of the FOI offices). We are writing to their police, visiting, providing substantial material proofs, since 2001, and it was always about the same gang of people attacking our basic rights. They only progress in severity of crime because the police refuses to serve us. That is apparently a cultural trait of AU. The rarest thing in the place is actually getting well served by the authorities for law and order. They seem to prefer killing, impairing, diseasing people criminally, and others, to assuming their mistakes before an originally overseas person, paying for them, and etc. They prefer claiming a citizen of theirs is crazy to claiming that they have unethical, unprofessional, incompetent, or criminal employees in their government. Examples of cases accumulate. We never stopped being told cases of other people who were told to be crazy because of that in the place. We feel shame of belonging to human kind since end of 2001, and it was never because of who or what we are... . Human kind deserves extermination as a whole. This is first world and they do what they do... .
19
Note on Gödel's first incompleteness theorem I. M. R. Pinheiro Abstract In this note, we nullify one of the most famous theorems used by Gödel to prove the incompleteness of the axiomatic systems for Arithmetic. Key-words: Gödel, P sentence, incompleteness, arithmetic, axiomatic system MSC2010: 03B05 Introduction Gödel has allegedly (scientific literature) created at least four proofs of the incompleteness of the axiomatic systems for Arithmetic: diagonalization that generates an unprovable sentence, the class K and the consequences of its existence, the sentence that asserts its own provability, and the ω-consistency theorem consequence. In [Juliette Kennedy, 2011], we read that the first incompleteness 1/19
Transcript
Note on Goumldels first incompleteness theorem
I M R Pinheiro
Abstract
In this note we nullify one of the most famous theorems used by Goumldel to prove the
incompleteness of the axiomatic systems for Arithmetic
Key-words Goumldel P sentence incompleteness arithmetic axiomatic system
MSC2010 03B05
Introduction
Goumldel has allegedly (scientific literature) created at least four proofs of the incompleteness of the
axiomatic systems for Arithmetic diagonalization that generates an unprovable sentence the class
K and the consequences of its existence the sentence that asserts its own provability and the ω-
consistency theorem consequence
In [Juliette Kennedy 2011] we read that the first incompleteness theorem of Goumldel is
ldquoIf P is ω-consistent then there is a sentence which is neither provable nor refutable from Prdquo
If the proof of this theorem were accepted Goumldel would have presented a system that contained an
unprovable sentence therefore a system that was incomplete and was contained in the usual
Arithmetic system
The theorem has been accepted as a theorem for Classical Logic because a proof that was
considered sound has been presented
However in the page 10 of [Kurt Goumldel 1930] we do not find material to scientifically convince
ourselves that ω-consistency has to do with consistency It seems that Goumldel simply asserted that
112
Every ω-consistent system is naturally also consistent in his work Despite his language being so
complex and hard to understand like his symbols and developments we dare stating that we cannot
see how such a thing could be true even after reading his writings from [Kurt Goumldel 1930]
Therefore there is a high chance that his incompleteness theorem is also not sound despite the
historical scientific choice of human kind
The other part of the mentioned work of Goumldel that seems to have generated the belief that
Arithmetic was not complete is about a class K that Goumldel invents in his writings supposed to be
formed from the elements that hold an ordering relation that cannot be proven
Such a thing could make sense because the ordering relation is created through axioms of definition
of the quantities for instance like the name we give to the quantities say 5 to five units are chosen
by us according to our taste and then assigned to a set containing those quantities via axiom that is
via an unprovable statement
However not many people even if chosen amongst those who are outside of the scientific world
would doubt that a set with five units of anything has to be bigger in some sense than a set with
four units of the same thing
Notice that he states that his K class is formed by natural numbers so that we know that we talk
about natural numbers for which the ordering relation cannot be proven
We here discuss this idea of Kurt Goumldel and actually argue that it does not generate any conflict
contrary to what he states in the mentioned work of his
According to him his class and some logical development of his would provide him with a
proposition that asserts its own provability
He compares his statements to The Liar and we have solved The Liar have recently put the solution
in writing so that we must be right when we state that his evidence despite coming in the shape of
at least four different reasoning lines is not actual evidence on the existence of sentences that are
valid in Arithmetic (use its lingo) and are unprovable inside of it
212
Some people and amongst them Juliette Kennedy [2011] assert that what Goumldel actually did was
using a statement this sentence is unprovable to prove the incompleteness of at least some of the
existing axiomatic systems (in particular the incompleteness of Arithmetic)
Considering the work of Goumldel that we talk about here the original work of Goumldel we disagree
with Juliette Kennedy We believe that he said something similar to that but not that
We believe that Goumldel was after the unprovable proposition with his writings from 1930 that we
here refer to instead like a proposition that could not be proven to be either false or true inside of
Arithmetic
That would then lead to the proof of the assertion Arithmetic is not complete as it was apparently
his intention
In the first page of his work that we here discuss from 1930 Goumldel actually brings a formula that
he calls F(v) that is a mathematical formula obviously then inside of Arithmetic that in his own
words (translated into English) would if interpreted as to content state v is a provable formula
Notice that F(v) is not the same as v yet the existing scientific literature seems to assert that Goumldel
has called F(v) v what can only be an enormous mistake in Science easily proven to be a mistake
(we cannot talk about a sentence that we do not know yet in full in a scientific manner)
Goumldels original F(v) does not generate any conflicting inferences as it is easy to see
The K-class example
According to our source which is a translation of the original works of Goumldel we find out that the
K-class of natural numbers is formed in the way that we exposed in this paper that is from the
natural numbers with ordering relation that cannot be proven
We will expose line by line of the reasoning of Kurt Goumldel in the source that we mention and will
produce remarks of quality for each one of those lines here
312
1st line Since the concepts that appear in the definiens are all definable in PM so too is the concept
K which is constituted from them ie there is a class-sign S such that the formula [Sn]-
interpreted as to its content- states that the natural number n belongs to K S being a class-sign is
identical with some determinate R(q) ie S = R(q)
Remarks We actually believe that it is probably the case that the best translation of what the
translator of the works of Goumldel calls class-sign is class representative because it is supposed to be
a formula of PM with just one free variable As an example Goumldel mentions [αn] which would be
the formula that is derived from replacing the free variable in the class representative α with the
representative for the natural number n In our heads we could have for instance α n + 2 and
[α3] 3 + 2 We believe that Goumldel called our modern mathematical concept function of a variable
some determinate q that is R(q) just meant a mathematical formula containing q as its only
variable or place holder q belonging to the natural numbers set
2nd line We now show that the proposition [R(q)q] is undecidable
Remarks The proposition [R(q)q] of Goumldel is the ordering relation R for the constant value
chosen from the natural numbers assigned to q Stating that such a proposition is undecidable
means saying that we cannot produce a proof of what we state about the order of the natural
numbers regarding the placement of that evaluation of q in the natural numbers line
3rd line Supposing that the proposition [R(q)q] were provable it would also be correct but that
means as has been said that q would belong to K
Remarks This line unfortunately frontally contradicts what Goumldel states himself in this work we
discuss here He actually said that K was formed by those instances of q with ordering relation that
could not be proven therefore q would not belong to K instead
4th line According to the definition of the class K we do not have a proof of [R(q)q] but we have
supposed that we had a proof for that
Remarks K states that we do not have a proof of [R(q)q] and we reached the conclusion that q
412
does not belong to K therefore we have a proof of [R(q)q] what is absolutely consistent with our
assumption
5th line If on the contrary the negation of [R(q)q] were provable then n would not belong to K
ie we would not have a proof of [R(q)q]
Remarks If we can prove that the assumed ordering relation for q is not true then we obviously do
not have a proof of [R(q)q] and therefore according to the definition of the K class by the own
Goumldel in the discussed work q would belong to K (not n as it appears in the translated text but q
instead obviously) precisely the opposite to what is asserted (again)
6th line [R(q)q] would thus be provable at the same time as its negation which again is
impossible
Remarks Trivially there are absolutely no conflicts instead because it is not true that [R(q)q] and
its negation are provable at the same time
Goumldel never found an example of mathematical formula that were undecidable inside of Arithmetic
through the K-class example therefore
The self-referential sentence
In large amount of the popular texts (see for instance [Kleene et al 1986]) we find Goumldel being
mentioned as if he had created a proposition that went like this V = V is unprovable in Г
The argument presented when such a possibility is exhibited is that if V is evaluated as true then V
is unprovable what then is a proof of incompleteness On the other hand if V is evaluated as false
then V must be provable but it states that it is unprovable so that we are left with a proposition that
we cannot judge in terms of truth-value which is what we wanted to achieve in order to defend the
incompleteness of the axiomatic systems that refer to the natural numbers
The problems with the possible proof of incompleteness that we have just presented are several The
512
most obvious of them all is perhaps what we have already mentioned There is a temporal obstacle
that cannot be removed Basically we will have V containing no determined sentence in V is
unprovable in Г but V will become determined as we name the sentence that we have just
mentioned V what then makes the example be rejected by Science for we cannot assert something
scientifically about something that we have not yet defined
To mention one more argument we have the problem with the detachment issue we present in our
analysis of The Liar Scientific statements must be completely cold and an entity asserting things
about themselves is not what we could call cold Instead that would be the warmest situation of all
of unavoidable attachment as in opposition to detachment and we insist on detachment
impartiality in all senses being an absolutely necessary condition for us to claim to be doing
Science
The ω-consistency theorem consequence
Two basic problems prevent this argument from being sound The first problem is what we have
already mentioned We cannot really find any evidence that ω-consistency implies consistency The
second is that the symbols of Goumldel created apparently to codify formulas in Mathematics (good
question would be why like are they not already codes and best as possible) like to replace
mathematical symbols that are used world-wide such as ( with numbers seem to confound the own
Goumldel in worse ways than they confound us
It is trivially the case that anything that may be done with a new set of symbols for mathematical
formulas may be done with the current set of symbols so why is it that Goumldel would be worried
about creating a new set of symbols
Goumldel is apparently trying to refer to things like generalization of statements with his symbols For
instance what we have mentioned earlier on Suppose that α n + 2 and [α3] 3 + 2 We then know
612
that α generalizes [α3] Mathematics has to appeal to natural language in order to describe what is
going on there to perfection is it not Goumldel was not happy with that He wanted symbols to replace
those words (generalizes)
We then start thinking that his objective was finding a formula inside of the metalanguage that were
unprovable right
That is what everyone says (P This sentence is not provable)
However Goumldel actually always worked with his own symbols in this development and never left
the original symbols of Mathematics (nothing is outside of Mathematics when translated) reaching
the conclusion that 17 Gen r is not c-provable instead
From this supposed finding he generalizes to v Gen r not being decidable
When we go through his symbols and try to work out what this so few symbols together mean the
disappointment cannot be avoided
We go translating his lingo using his list of symbols and reach a term that cannot be translated by
step six of our translation processes which is n St 17 = 0 (we are presented with item 28 k St v x
but we see no sign of anything like k St v)
If we cannot find a way of translating this using his own symbols then it is easy to infer that we do
not have an example of an unprovable formula
Besides there is nothing that he writes with his symbols that cannot be translated into our usual
mathematical lingo and usual natural language symbols If he really had an example of such a
sentence why would he not present that to us even if at the end of his deductions in the simplest
way as possible
Without going into the merit of his writings for for instance the translator or even the assistant of
the translator could have swapped = 0 for something else in the remark of the item 28 we know that
what is being presented to us is not being presented to standards In Science communication is
obviously priority Many researchers have spent an entire life perfecting mathematical lingo for
712
instance so that everything would be as objective and universal as possible If something can be
passed onwards in a simpler form and we know that we have obligation of doing so if claiming to
be writing Science
Diagonalization that generates an unprovable sentence
From the writings that we had access to in the past not the original writings of Goumldel but writings
of other authors that wrote about his writings we understood that Goumldel simply used his codes to
make a list that were similar to the one presented by Cantor to prove that the real numbers
considered as a whole were uncountable
Cantor (see for instance [Eric Weisstein 1999]) would have tried to make a correspondence
between natural numbers and the real numbers and would have ended up proving that such was
impossible through forming a new number which was not part of his original list by picking one
digit from each position that corresponded to a diagonal line traced from the first number to the last
number he had put in the list and making his new number differ from that digit in that position
If his new number differs from each number in the list by the nth decimal digit where nth
corresponds to the ordinal number representing the listed number in his original list say then the
new number cannot be equal to any of the numbers previously listed by him
The work of diagonalization of Cantor which is what we have just mentioned seems unnecessary
to prove such an easy-to-understand claim What is being claimed is that we cannot assign ordinal
numbers (see [Eric Weisstein 1999b] for instance) in bijection or one-to-one correspondence to
any non-degenerated slice of the real numbers line and therefore we cannot count considering
how Mathematics has defined the operation of counting the real numbers not even in a small
interval like even in the smallest as possible interval that be non-degenerated
We could have proven that by simply listing numbers of the shape 0something as Cantor does but
812
with first decimal digit being the own natural number that we wish to use to count
It is easy to see that there are plenty of reals in between each two lines of the just-assembled list so
that the diagonal formation is unnecessary
In any hypothesis it is claimed that Goumldel used this diagonalization of Cantor to once more
exhibit an unprovable statement from the Arithmetic World
He would then have used his symbols and would have listed all possible formula of the type a + b =
c lets say
Obviously the case that the assignments of value to a b and c will use the entire spectrum of the
natural numbers so that varying only one of those variables in the formula already has covered the
ordinal numbers
Utilizing Cantors argument here means creating a formula that is not listed and claiming that we do
not have an ordinal for it
However Goumldel changes that into the formula not being passive of deduction from the previous
formulas in the list because it will differ from each one of those formulas in the list by the n th digit
where nth is the order in the list of each one of those formulas
Thinking about it leads us to doubt his statement straight away For instance the operation of
multiplication derives from the operation of summing We could have several lines of summing and
infer multiplication The only symbol in common would be the equal sign
2 x 12 = 24 comes from perhaps 2412 = 2
Considering a b and c here plus codes for the operations would lead us to having only the sign =
and the number 12 as common elements Yet one formula did come from the other
To eliminate the equal sign we can think of an example from Set Theory for instance
From 1 S a є S b є S 2 S c є S d є Snotin notin until 13 S y є S z є Snotin plus X = a b c
hellip z we infer X C S
Now if we consider the position of the elements in the line and symbols we will have no
912
intersection between those lines and the conclusion Yet we have no doubts that the conclusion does
derive from the premises
We have not seen the original works of Goumldel only had access to what other people have written
about his work on this proof but we understand that he either commits a mistake of the order that
we have just pointed or he forgets that the formula has to make sense for sometimes he is told to
have simply used all his symbols chaotically and produced a formula from a diagonal over the list at
the end
The main problem with this proof is the story of coding what is already coded basically
As said before his coding should only be considered scientifically acceptable if it could create a
formula that is unprovable inside of the metalanguage for Mathematics because we should always
go in the simplest way as possible in scientific argumentation and if his example involves what
could have been described with the standard codessymbols for Mathematics then it should have
been obviously
Conclusion
There is no actual scientific evidence on the incompleteness of axiomatic systems for Arithmetic
coming from the works of Goumldel therefore we must assume that the axiomatic system that we use
for Arithmetic is complete
1012
References
Eric Weisstein (1999) Cantor Diagonal Method Retrieved 14 November 2011
from MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCantorDiagonalMethodhtml
Eric Weisstein (1999b) Countable Set Retrieved 14 November 2011 from
MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCountableSethtml
Juliette Kennedy (2011) Kurt Goumldel The Stanford Encyclopedia of Philosophy (Fall 2011
Edition) Edward N Zalta (ed) Retrieved November 10 2011 from
httpplatostanfordeduarchivesfall2011entriesgoedel
Kurt Goumldel (1930) On Formally undecidable propositions of Principia Mathematica and related
systems 1 Yggdrasils WN Library Retrieved 10 November 2011 from
httpwwwgeierhuGOEDELGoumldel_origGoumldel3htm
Robert Constable (2009) Lecture 23 Unsolvable Problems in Logic Applied Logic course CS
4860 Department of Computer Science Cornell University Retrieved 14 November 2011 from
httpwwwcscornelleducoursescs48602009splec-23pdf
1112
S Feferman J D W Junior S C Kleene G H Moore R M Soloway and J V
Heijenoort (eds) (1986) Kurt Goumldel Collected Works volume I publications
1929-1936 Oxford University Press ISBN-13 978-0195039641
Postal address for the author
P O Box 12396 ABeckett st Melbourne Victoria Australia 3000
1212
Every ω-consistent system is naturally also consistent in his work Despite his language being so
complex and hard to understand like his symbols and developments we dare stating that we cannot
see how such a thing could be true even after reading his writings from [Kurt Goumldel 1930]
Therefore there is a high chance that his incompleteness theorem is also not sound despite the
historical scientific choice of human kind
The other part of the mentioned work of Goumldel that seems to have generated the belief that
Arithmetic was not complete is about a class K that Goumldel invents in his writings supposed to be
formed from the elements that hold an ordering relation that cannot be proven
Such a thing could make sense because the ordering relation is created through axioms of definition
of the quantities for instance like the name we give to the quantities say 5 to five units are chosen
by us according to our taste and then assigned to a set containing those quantities via axiom that is
via an unprovable statement
However not many people even if chosen amongst those who are outside of the scientific world
would doubt that a set with five units of anything has to be bigger in some sense than a set with
four units of the same thing
Notice that he states that his K class is formed by natural numbers so that we know that we talk
about natural numbers for which the ordering relation cannot be proven
We here discuss this idea of Kurt Goumldel and actually argue that it does not generate any conflict
contrary to what he states in the mentioned work of his
According to him his class and some logical development of his would provide him with a
proposition that asserts its own provability
He compares his statements to The Liar and we have solved The Liar have recently put the solution
in writing so that we must be right when we state that his evidence despite coming in the shape of
at least four different reasoning lines is not actual evidence on the existence of sentences that are
valid in Arithmetic (use its lingo) and are unprovable inside of it
212
Some people and amongst them Juliette Kennedy [2011] assert that what Goumldel actually did was
using a statement this sentence is unprovable to prove the incompleteness of at least some of the
existing axiomatic systems (in particular the incompleteness of Arithmetic)
Considering the work of Goumldel that we talk about here the original work of Goumldel we disagree
with Juliette Kennedy We believe that he said something similar to that but not that
We believe that Goumldel was after the unprovable proposition with his writings from 1930 that we
here refer to instead like a proposition that could not be proven to be either false or true inside of
Arithmetic
That would then lead to the proof of the assertion Arithmetic is not complete as it was apparently
his intention
In the first page of his work that we here discuss from 1930 Goumldel actually brings a formula that
he calls F(v) that is a mathematical formula obviously then inside of Arithmetic that in his own
words (translated into English) would if interpreted as to content state v is a provable formula
Notice that F(v) is not the same as v yet the existing scientific literature seems to assert that Goumldel
has called F(v) v what can only be an enormous mistake in Science easily proven to be a mistake
(we cannot talk about a sentence that we do not know yet in full in a scientific manner)
Goumldels original F(v) does not generate any conflicting inferences as it is easy to see
The K-class example
According to our source which is a translation of the original works of Goumldel we find out that the
K-class of natural numbers is formed in the way that we exposed in this paper that is from the
natural numbers with ordering relation that cannot be proven
We will expose line by line of the reasoning of Kurt Goumldel in the source that we mention and will
produce remarks of quality for each one of those lines here
312
1st line Since the concepts that appear in the definiens are all definable in PM so too is the concept
K which is constituted from them ie there is a class-sign S such that the formula [Sn]-
interpreted as to its content- states that the natural number n belongs to K S being a class-sign is
identical with some determinate R(q) ie S = R(q)
Remarks We actually believe that it is probably the case that the best translation of what the
translator of the works of Goumldel calls class-sign is class representative because it is supposed to be
a formula of PM with just one free variable As an example Goumldel mentions [αn] which would be
the formula that is derived from replacing the free variable in the class representative α with the
representative for the natural number n In our heads we could have for instance α n + 2 and
[α3] 3 + 2 We believe that Goumldel called our modern mathematical concept function of a variable
some determinate q that is R(q) just meant a mathematical formula containing q as its only
variable or place holder q belonging to the natural numbers set
2nd line We now show that the proposition [R(q)q] is undecidable
Remarks The proposition [R(q)q] of Goumldel is the ordering relation R for the constant value
chosen from the natural numbers assigned to q Stating that such a proposition is undecidable
means saying that we cannot produce a proof of what we state about the order of the natural
numbers regarding the placement of that evaluation of q in the natural numbers line
3rd line Supposing that the proposition [R(q)q] were provable it would also be correct but that
means as has been said that q would belong to K
Remarks This line unfortunately frontally contradicts what Goumldel states himself in this work we
discuss here He actually said that K was formed by those instances of q with ordering relation that
could not be proven therefore q would not belong to K instead
4th line According to the definition of the class K we do not have a proof of [R(q)q] but we have
supposed that we had a proof for that
Remarks K states that we do not have a proof of [R(q)q] and we reached the conclusion that q
412
does not belong to K therefore we have a proof of [R(q)q] what is absolutely consistent with our
assumption
5th line If on the contrary the negation of [R(q)q] were provable then n would not belong to K
ie we would not have a proof of [R(q)q]
Remarks If we can prove that the assumed ordering relation for q is not true then we obviously do
not have a proof of [R(q)q] and therefore according to the definition of the K class by the own
Goumldel in the discussed work q would belong to K (not n as it appears in the translated text but q
instead obviously) precisely the opposite to what is asserted (again)
6th line [R(q)q] would thus be provable at the same time as its negation which again is
impossible
Remarks Trivially there are absolutely no conflicts instead because it is not true that [R(q)q] and
its negation are provable at the same time
Goumldel never found an example of mathematical formula that were undecidable inside of Arithmetic
through the K-class example therefore
The self-referential sentence
In large amount of the popular texts (see for instance [Kleene et al 1986]) we find Goumldel being
mentioned as if he had created a proposition that went like this V = V is unprovable in Г
The argument presented when such a possibility is exhibited is that if V is evaluated as true then V
is unprovable what then is a proof of incompleteness On the other hand if V is evaluated as false
then V must be provable but it states that it is unprovable so that we are left with a proposition that
we cannot judge in terms of truth-value which is what we wanted to achieve in order to defend the
incompleteness of the axiomatic systems that refer to the natural numbers
The problems with the possible proof of incompleteness that we have just presented are several The
512
most obvious of them all is perhaps what we have already mentioned There is a temporal obstacle
that cannot be removed Basically we will have V containing no determined sentence in V is
unprovable in Г but V will become determined as we name the sentence that we have just
mentioned V what then makes the example be rejected by Science for we cannot assert something
scientifically about something that we have not yet defined
To mention one more argument we have the problem with the detachment issue we present in our
analysis of The Liar Scientific statements must be completely cold and an entity asserting things
about themselves is not what we could call cold Instead that would be the warmest situation of all
of unavoidable attachment as in opposition to detachment and we insist on detachment
impartiality in all senses being an absolutely necessary condition for us to claim to be doing
Science
The ω-consistency theorem consequence
Two basic problems prevent this argument from being sound The first problem is what we have
already mentioned We cannot really find any evidence that ω-consistency implies consistency The
second is that the symbols of Goumldel created apparently to codify formulas in Mathematics (good
question would be why like are they not already codes and best as possible) like to replace
mathematical symbols that are used world-wide such as ( with numbers seem to confound the own
Goumldel in worse ways than they confound us
It is trivially the case that anything that may be done with a new set of symbols for mathematical
formulas may be done with the current set of symbols so why is it that Goumldel would be worried
about creating a new set of symbols
Goumldel is apparently trying to refer to things like generalization of statements with his symbols For
instance what we have mentioned earlier on Suppose that α n + 2 and [α3] 3 + 2 We then know
612
that α generalizes [α3] Mathematics has to appeal to natural language in order to describe what is
going on there to perfection is it not Goumldel was not happy with that He wanted symbols to replace
those words (generalizes)
We then start thinking that his objective was finding a formula inside of the metalanguage that were
unprovable right
That is what everyone says (P This sentence is not provable)
However Goumldel actually always worked with his own symbols in this development and never left
the original symbols of Mathematics (nothing is outside of Mathematics when translated) reaching
the conclusion that 17 Gen r is not c-provable instead
From this supposed finding he generalizes to v Gen r not being decidable
When we go through his symbols and try to work out what this so few symbols together mean the
disappointment cannot be avoided
We go translating his lingo using his list of symbols and reach a term that cannot be translated by
step six of our translation processes which is n St 17 = 0 (we are presented with item 28 k St v x
but we see no sign of anything like k St v)
If we cannot find a way of translating this using his own symbols then it is easy to infer that we do
not have an example of an unprovable formula
Besides there is nothing that he writes with his symbols that cannot be translated into our usual
mathematical lingo and usual natural language symbols If he really had an example of such a
sentence why would he not present that to us even if at the end of his deductions in the simplest
way as possible
Without going into the merit of his writings for for instance the translator or even the assistant of
the translator could have swapped = 0 for something else in the remark of the item 28 we know that
what is being presented to us is not being presented to standards In Science communication is
obviously priority Many researchers have spent an entire life perfecting mathematical lingo for
712
instance so that everything would be as objective and universal as possible If something can be
passed onwards in a simpler form and we know that we have obligation of doing so if claiming to
be writing Science
Diagonalization that generates an unprovable sentence
From the writings that we had access to in the past not the original writings of Goumldel but writings
of other authors that wrote about his writings we understood that Goumldel simply used his codes to
make a list that were similar to the one presented by Cantor to prove that the real numbers
considered as a whole were uncountable
Cantor (see for instance [Eric Weisstein 1999]) would have tried to make a correspondence
between natural numbers and the real numbers and would have ended up proving that such was
impossible through forming a new number which was not part of his original list by picking one
digit from each position that corresponded to a diagonal line traced from the first number to the last
number he had put in the list and making his new number differ from that digit in that position
If his new number differs from each number in the list by the nth decimal digit where nth
corresponds to the ordinal number representing the listed number in his original list say then the
new number cannot be equal to any of the numbers previously listed by him
The work of diagonalization of Cantor which is what we have just mentioned seems unnecessary
to prove such an easy-to-understand claim What is being claimed is that we cannot assign ordinal
numbers (see [Eric Weisstein 1999b] for instance) in bijection or one-to-one correspondence to
any non-degenerated slice of the real numbers line and therefore we cannot count considering
how Mathematics has defined the operation of counting the real numbers not even in a small
interval like even in the smallest as possible interval that be non-degenerated
We could have proven that by simply listing numbers of the shape 0something as Cantor does but
812
with first decimal digit being the own natural number that we wish to use to count
It is easy to see that there are plenty of reals in between each two lines of the just-assembled list so
that the diagonal formation is unnecessary
In any hypothesis it is claimed that Goumldel used this diagonalization of Cantor to once more
exhibit an unprovable statement from the Arithmetic World
He would then have used his symbols and would have listed all possible formula of the type a + b =
c lets say
Obviously the case that the assignments of value to a b and c will use the entire spectrum of the
natural numbers so that varying only one of those variables in the formula already has covered the
ordinal numbers
Utilizing Cantors argument here means creating a formula that is not listed and claiming that we do
not have an ordinal for it
However Goumldel changes that into the formula not being passive of deduction from the previous
formulas in the list because it will differ from each one of those formulas in the list by the n th digit
where nth is the order in the list of each one of those formulas
Thinking about it leads us to doubt his statement straight away For instance the operation of
multiplication derives from the operation of summing We could have several lines of summing and
infer multiplication The only symbol in common would be the equal sign
2 x 12 = 24 comes from perhaps 2412 = 2
Considering a b and c here plus codes for the operations would lead us to having only the sign =
and the number 12 as common elements Yet one formula did come from the other
To eliminate the equal sign we can think of an example from Set Theory for instance
From 1 S a є S b є S 2 S c є S d є Snotin notin until 13 S y є S z є Snotin plus X = a b c
hellip z we infer X C S
Now if we consider the position of the elements in the line and symbols we will have no
912
intersection between those lines and the conclusion Yet we have no doubts that the conclusion does
derive from the premises
We have not seen the original works of Goumldel only had access to what other people have written
about his work on this proof but we understand that he either commits a mistake of the order that
we have just pointed or he forgets that the formula has to make sense for sometimes he is told to
have simply used all his symbols chaotically and produced a formula from a diagonal over the list at
the end
The main problem with this proof is the story of coding what is already coded basically
As said before his coding should only be considered scientifically acceptable if it could create a
formula that is unprovable inside of the metalanguage for Mathematics because we should always
go in the simplest way as possible in scientific argumentation and if his example involves what
could have been described with the standard codessymbols for Mathematics then it should have
been obviously
Conclusion
There is no actual scientific evidence on the incompleteness of axiomatic systems for Arithmetic
coming from the works of Goumldel therefore we must assume that the axiomatic system that we use
for Arithmetic is complete
1012
References
Eric Weisstein (1999) Cantor Diagonal Method Retrieved 14 November 2011
from MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCantorDiagonalMethodhtml
Eric Weisstein (1999b) Countable Set Retrieved 14 November 2011 from
MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCountableSethtml
Juliette Kennedy (2011) Kurt Goumldel The Stanford Encyclopedia of Philosophy (Fall 2011
Edition) Edward N Zalta (ed) Retrieved November 10 2011 from
httpplatostanfordeduarchivesfall2011entriesgoedel
Kurt Goumldel (1930) On Formally undecidable propositions of Principia Mathematica and related
systems 1 Yggdrasils WN Library Retrieved 10 November 2011 from
httpwwwgeierhuGOEDELGoumldel_origGoumldel3htm
Robert Constable (2009) Lecture 23 Unsolvable Problems in Logic Applied Logic course CS
4860 Department of Computer Science Cornell University Retrieved 14 November 2011 from
httpwwwcscornelleducoursescs48602009splec-23pdf
1112
S Feferman J D W Junior S C Kleene G H Moore R M Soloway and J V
Heijenoort (eds) (1986) Kurt Goumldel Collected Works volume I publications
1929-1936 Oxford University Press ISBN-13 978-0195039641
Postal address for the author
P O Box 12396 ABeckett st Melbourne Victoria Australia 3000
1212
Some people and amongst them Juliette Kennedy [2011] assert that what Goumldel actually did was
using a statement this sentence is unprovable to prove the incompleteness of at least some of the
existing axiomatic systems (in particular the incompleteness of Arithmetic)
Considering the work of Goumldel that we talk about here the original work of Goumldel we disagree
with Juliette Kennedy We believe that he said something similar to that but not that
We believe that Goumldel was after the unprovable proposition with his writings from 1930 that we
here refer to instead like a proposition that could not be proven to be either false or true inside of
Arithmetic
That would then lead to the proof of the assertion Arithmetic is not complete as it was apparently
his intention
In the first page of his work that we here discuss from 1930 Goumldel actually brings a formula that
he calls F(v) that is a mathematical formula obviously then inside of Arithmetic that in his own
words (translated into English) would if interpreted as to content state v is a provable formula
Notice that F(v) is not the same as v yet the existing scientific literature seems to assert that Goumldel
has called F(v) v what can only be an enormous mistake in Science easily proven to be a mistake
(we cannot talk about a sentence that we do not know yet in full in a scientific manner)
Goumldels original F(v) does not generate any conflicting inferences as it is easy to see
The K-class example
According to our source which is a translation of the original works of Goumldel we find out that the
K-class of natural numbers is formed in the way that we exposed in this paper that is from the
natural numbers with ordering relation that cannot be proven
We will expose line by line of the reasoning of Kurt Goumldel in the source that we mention and will
produce remarks of quality for each one of those lines here
312
1st line Since the concepts that appear in the definiens are all definable in PM so too is the concept
K which is constituted from them ie there is a class-sign S such that the formula [Sn]-
interpreted as to its content- states that the natural number n belongs to K S being a class-sign is
identical with some determinate R(q) ie S = R(q)
Remarks We actually believe that it is probably the case that the best translation of what the
translator of the works of Goumldel calls class-sign is class representative because it is supposed to be
a formula of PM with just one free variable As an example Goumldel mentions [αn] which would be
the formula that is derived from replacing the free variable in the class representative α with the
representative for the natural number n In our heads we could have for instance α n + 2 and
[α3] 3 + 2 We believe that Goumldel called our modern mathematical concept function of a variable
some determinate q that is R(q) just meant a mathematical formula containing q as its only
variable or place holder q belonging to the natural numbers set
2nd line We now show that the proposition [R(q)q] is undecidable
Remarks The proposition [R(q)q] of Goumldel is the ordering relation R for the constant value
chosen from the natural numbers assigned to q Stating that such a proposition is undecidable
means saying that we cannot produce a proof of what we state about the order of the natural
numbers regarding the placement of that evaluation of q in the natural numbers line
3rd line Supposing that the proposition [R(q)q] were provable it would also be correct but that
means as has been said that q would belong to K
Remarks This line unfortunately frontally contradicts what Goumldel states himself in this work we
discuss here He actually said that K was formed by those instances of q with ordering relation that
could not be proven therefore q would not belong to K instead
4th line According to the definition of the class K we do not have a proof of [R(q)q] but we have
supposed that we had a proof for that
Remarks K states that we do not have a proof of [R(q)q] and we reached the conclusion that q
412
does not belong to K therefore we have a proof of [R(q)q] what is absolutely consistent with our
assumption
5th line If on the contrary the negation of [R(q)q] were provable then n would not belong to K
ie we would not have a proof of [R(q)q]
Remarks If we can prove that the assumed ordering relation for q is not true then we obviously do
not have a proof of [R(q)q] and therefore according to the definition of the K class by the own
Goumldel in the discussed work q would belong to K (not n as it appears in the translated text but q
instead obviously) precisely the opposite to what is asserted (again)
6th line [R(q)q] would thus be provable at the same time as its negation which again is
impossible
Remarks Trivially there are absolutely no conflicts instead because it is not true that [R(q)q] and
its negation are provable at the same time
Goumldel never found an example of mathematical formula that were undecidable inside of Arithmetic
through the K-class example therefore
The self-referential sentence
In large amount of the popular texts (see for instance [Kleene et al 1986]) we find Goumldel being
mentioned as if he had created a proposition that went like this V = V is unprovable in Г
The argument presented when such a possibility is exhibited is that if V is evaluated as true then V
is unprovable what then is a proof of incompleteness On the other hand if V is evaluated as false
then V must be provable but it states that it is unprovable so that we are left with a proposition that
we cannot judge in terms of truth-value which is what we wanted to achieve in order to defend the
incompleteness of the axiomatic systems that refer to the natural numbers
The problems with the possible proof of incompleteness that we have just presented are several The
512
most obvious of them all is perhaps what we have already mentioned There is a temporal obstacle
that cannot be removed Basically we will have V containing no determined sentence in V is
unprovable in Г but V will become determined as we name the sentence that we have just
mentioned V what then makes the example be rejected by Science for we cannot assert something
scientifically about something that we have not yet defined
To mention one more argument we have the problem with the detachment issue we present in our
analysis of The Liar Scientific statements must be completely cold and an entity asserting things
about themselves is not what we could call cold Instead that would be the warmest situation of all
of unavoidable attachment as in opposition to detachment and we insist on detachment
impartiality in all senses being an absolutely necessary condition for us to claim to be doing
Science
The ω-consistency theorem consequence
Two basic problems prevent this argument from being sound The first problem is what we have
already mentioned We cannot really find any evidence that ω-consistency implies consistency The
second is that the symbols of Goumldel created apparently to codify formulas in Mathematics (good
question would be why like are they not already codes and best as possible) like to replace
mathematical symbols that are used world-wide such as ( with numbers seem to confound the own
Goumldel in worse ways than they confound us
It is trivially the case that anything that may be done with a new set of symbols for mathematical
formulas may be done with the current set of symbols so why is it that Goumldel would be worried
about creating a new set of symbols
Goumldel is apparently trying to refer to things like generalization of statements with his symbols For
instance what we have mentioned earlier on Suppose that α n + 2 and [α3] 3 + 2 We then know
612
that α generalizes [α3] Mathematics has to appeal to natural language in order to describe what is
going on there to perfection is it not Goumldel was not happy with that He wanted symbols to replace
those words (generalizes)
We then start thinking that his objective was finding a formula inside of the metalanguage that were
unprovable right
That is what everyone says (P This sentence is not provable)
However Goumldel actually always worked with his own symbols in this development and never left
the original symbols of Mathematics (nothing is outside of Mathematics when translated) reaching
the conclusion that 17 Gen r is not c-provable instead
From this supposed finding he generalizes to v Gen r not being decidable
When we go through his symbols and try to work out what this so few symbols together mean the
disappointment cannot be avoided
We go translating his lingo using his list of symbols and reach a term that cannot be translated by
step six of our translation processes which is n St 17 = 0 (we are presented with item 28 k St v x
but we see no sign of anything like k St v)
If we cannot find a way of translating this using his own symbols then it is easy to infer that we do
not have an example of an unprovable formula
Besides there is nothing that he writes with his symbols that cannot be translated into our usual
mathematical lingo and usual natural language symbols If he really had an example of such a
sentence why would he not present that to us even if at the end of his deductions in the simplest
way as possible
Without going into the merit of his writings for for instance the translator or even the assistant of
the translator could have swapped = 0 for something else in the remark of the item 28 we know that
what is being presented to us is not being presented to standards In Science communication is
obviously priority Many researchers have spent an entire life perfecting mathematical lingo for
712
instance so that everything would be as objective and universal as possible If something can be
passed onwards in a simpler form and we know that we have obligation of doing so if claiming to
be writing Science
Diagonalization that generates an unprovable sentence
From the writings that we had access to in the past not the original writings of Goumldel but writings
of other authors that wrote about his writings we understood that Goumldel simply used his codes to
make a list that were similar to the one presented by Cantor to prove that the real numbers
considered as a whole were uncountable
Cantor (see for instance [Eric Weisstein 1999]) would have tried to make a correspondence
between natural numbers and the real numbers and would have ended up proving that such was
impossible through forming a new number which was not part of his original list by picking one
digit from each position that corresponded to a diagonal line traced from the first number to the last
number he had put in the list and making his new number differ from that digit in that position
If his new number differs from each number in the list by the nth decimal digit where nth
corresponds to the ordinal number representing the listed number in his original list say then the
new number cannot be equal to any of the numbers previously listed by him
The work of diagonalization of Cantor which is what we have just mentioned seems unnecessary
to prove such an easy-to-understand claim What is being claimed is that we cannot assign ordinal
numbers (see [Eric Weisstein 1999b] for instance) in bijection or one-to-one correspondence to
any non-degenerated slice of the real numbers line and therefore we cannot count considering
how Mathematics has defined the operation of counting the real numbers not even in a small
interval like even in the smallest as possible interval that be non-degenerated
We could have proven that by simply listing numbers of the shape 0something as Cantor does but
812
with first decimal digit being the own natural number that we wish to use to count
It is easy to see that there are plenty of reals in between each two lines of the just-assembled list so
that the diagonal formation is unnecessary
In any hypothesis it is claimed that Goumldel used this diagonalization of Cantor to once more
exhibit an unprovable statement from the Arithmetic World
He would then have used his symbols and would have listed all possible formula of the type a + b =
c lets say
Obviously the case that the assignments of value to a b and c will use the entire spectrum of the
natural numbers so that varying only one of those variables in the formula already has covered the
ordinal numbers
Utilizing Cantors argument here means creating a formula that is not listed and claiming that we do
not have an ordinal for it
However Goumldel changes that into the formula not being passive of deduction from the previous
formulas in the list because it will differ from each one of those formulas in the list by the n th digit
where nth is the order in the list of each one of those formulas
Thinking about it leads us to doubt his statement straight away For instance the operation of
multiplication derives from the operation of summing We could have several lines of summing and
infer multiplication The only symbol in common would be the equal sign
2 x 12 = 24 comes from perhaps 2412 = 2
Considering a b and c here plus codes for the operations would lead us to having only the sign =
and the number 12 as common elements Yet one formula did come from the other
To eliminate the equal sign we can think of an example from Set Theory for instance
From 1 S a є S b є S 2 S c є S d є Snotin notin until 13 S y є S z є Snotin plus X = a b c
hellip z we infer X C S
Now if we consider the position of the elements in the line and symbols we will have no
912
intersection between those lines and the conclusion Yet we have no doubts that the conclusion does
derive from the premises
We have not seen the original works of Goumldel only had access to what other people have written
about his work on this proof but we understand that he either commits a mistake of the order that
we have just pointed or he forgets that the formula has to make sense for sometimes he is told to
have simply used all his symbols chaotically and produced a formula from a diagonal over the list at
the end
The main problem with this proof is the story of coding what is already coded basically
As said before his coding should only be considered scientifically acceptable if it could create a
formula that is unprovable inside of the metalanguage for Mathematics because we should always
go in the simplest way as possible in scientific argumentation and if his example involves what
could have been described with the standard codessymbols for Mathematics then it should have
been obviously
Conclusion
There is no actual scientific evidence on the incompleteness of axiomatic systems for Arithmetic
coming from the works of Goumldel therefore we must assume that the axiomatic system that we use
for Arithmetic is complete
1012
References
Eric Weisstein (1999) Cantor Diagonal Method Retrieved 14 November 2011
from MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCantorDiagonalMethodhtml
Eric Weisstein (1999b) Countable Set Retrieved 14 November 2011 from
MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCountableSethtml
Juliette Kennedy (2011) Kurt Goumldel The Stanford Encyclopedia of Philosophy (Fall 2011
Edition) Edward N Zalta (ed) Retrieved November 10 2011 from
httpplatostanfordeduarchivesfall2011entriesgoedel
Kurt Goumldel (1930) On Formally undecidable propositions of Principia Mathematica and related
systems 1 Yggdrasils WN Library Retrieved 10 November 2011 from
httpwwwgeierhuGOEDELGoumldel_origGoumldel3htm
Robert Constable (2009) Lecture 23 Unsolvable Problems in Logic Applied Logic course CS
4860 Department of Computer Science Cornell University Retrieved 14 November 2011 from
httpwwwcscornelleducoursescs48602009splec-23pdf
1112
S Feferman J D W Junior S C Kleene G H Moore R M Soloway and J V
Heijenoort (eds) (1986) Kurt Goumldel Collected Works volume I publications
1929-1936 Oxford University Press ISBN-13 978-0195039641
Postal address for the author
P O Box 12396 ABeckett st Melbourne Victoria Australia 3000
1212
1st line Since the concepts that appear in the definiens are all definable in PM so too is the concept
K which is constituted from them ie there is a class-sign S such that the formula [Sn]-
interpreted as to its content- states that the natural number n belongs to K S being a class-sign is
identical with some determinate R(q) ie S = R(q)
Remarks We actually believe that it is probably the case that the best translation of what the
translator of the works of Goumldel calls class-sign is class representative because it is supposed to be
a formula of PM with just one free variable As an example Goumldel mentions [αn] which would be
the formula that is derived from replacing the free variable in the class representative α with the
representative for the natural number n In our heads we could have for instance α n + 2 and
[α3] 3 + 2 We believe that Goumldel called our modern mathematical concept function of a variable
some determinate q that is R(q) just meant a mathematical formula containing q as its only
variable or place holder q belonging to the natural numbers set
2nd line We now show that the proposition [R(q)q] is undecidable
Remarks The proposition [R(q)q] of Goumldel is the ordering relation R for the constant value
chosen from the natural numbers assigned to q Stating that such a proposition is undecidable
means saying that we cannot produce a proof of what we state about the order of the natural
numbers regarding the placement of that evaluation of q in the natural numbers line
3rd line Supposing that the proposition [R(q)q] were provable it would also be correct but that
means as has been said that q would belong to K
Remarks This line unfortunately frontally contradicts what Goumldel states himself in this work we
discuss here He actually said that K was formed by those instances of q with ordering relation that
could not be proven therefore q would not belong to K instead
4th line According to the definition of the class K we do not have a proof of [R(q)q] but we have
supposed that we had a proof for that
Remarks K states that we do not have a proof of [R(q)q] and we reached the conclusion that q
412
does not belong to K therefore we have a proof of [R(q)q] what is absolutely consistent with our
assumption
5th line If on the contrary the negation of [R(q)q] were provable then n would not belong to K
ie we would not have a proof of [R(q)q]
Remarks If we can prove that the assumed ordering relation for q is not true then we obviously do
not have a proof of [R(q)q] and therefore according to the definition of the K class by the own
Goumldel in the discussed work q would belong to K (not n as it appears in the translated text but q
instead obviously) precisely the opposite to what is asserted (again)
6th line [R(q)q] would thus be provable at the same time as its negation which again is
impossible
Remarks Trivially there are absolutely no conflicts instead because it is not true that [R(q)q] and
its negation are provable at the same time
Goumldel never found an example of mathematical formula that were undecidable inside of Arithmetic
through the K-class example therefore
The self-referential sentence
In large amount of the popular texts (see for instance [Kleene et al 1986]) we find Goumldel being
mentioned as if he had created a proposition that went like this V = V is unprovable in Г
The argument presented when such a possibility is exhibited is that if V is evaluated as true then V
is unprovable what then is a proof of incompleteness On the other hand if V is evaluated as false
then V must be provable but it states that it is unprovable so that we are left with a proposition that
we cannot judge in terms of truth-value which is what we wanted to achieve in order to defend the
incompleteness of the axiomatic systems that refer to the natural numbers
The problems with the possible proof of incompleteness that we have just presented are several The
512
most obvious of them all is perhaps what we have already mentioned There is a temporal obstacle
that cannot be removed Basically we will have V containing no determined sentence in V is
unprovable in Г but V will become determined as we name the sentence that we have just
mentioned V what then makes the example be rejected by Science for we cannot assert something
scientifically about something that we have not yet defined
To mention one more argument we have the problem with the detachment issue we present in our
analysis of The Liar Scientific statements must be completely cold and an entity asserting things
about themselves is not what we could call cold Instead that would be the warmest situation of all
of unavoidable attachment as in opposition to detachment and we insist on detachment
impartiality in all senses being an absolutely necessary condition for us to claim to be doing
Science
The ω-consistency theorem consequence
Two basic problems prevent this argument from being sound The first problem is what we have
already mentioned We cannot really find any evidence that ω-consistency implies consistency The
second is that the symbols of Goumldel created apparently to codify formulas in Mathematics (good
question would be why like are they not already codes and best as possible) like to replace
mathematical symbols that are used world-wide such as ( with numbers seem to confound the own
Goumldel in worse ways than they confound us
It is trivially the case that anything that may be done with a new set of symbols for mathematical
formulas may be done with the current set of symbols so why is it that Goumldel would be worried
about creating a new set of symbols
Goumldel is apparently trying to refer to things like generalization of statements with his symbols For
instance what we have mentioned earlier on Suppose that α n + 2 and [α3] 3 + 2 We then know
612
that α generalizes [α3] Mathematics has to appeal to natural language in order to describe what is
going on there to perfection is it not Goumldel was not happy with that He wanted symbols to replace
those words (generalizes)
We then start thinking that his objective was finding a formula inside of the metalanguage that were
unprovable right
That is what everyone says (P This sentence is not provable)
However Goumldel actually always worked with his own symbols in this development and never left
the original symbols of Mathematics (nothing is outside of Mathematics when translated) reaching
the conclusion that 17 Gen r is not c-provable instead
From this supposed finding he generalizes to v Gen r not being decidable
When we go through his symbols and try to work out what this so few symbols together mean the
disappointment cannot be avoided
We go translating his lingo using his list of symbols and reach a term that cannot be translated by
step six of our translation processes which is n St 17 = 0 (we are presented with item 28 k St v x
but we see no sign of anything like k St v)
If we cannot find a way of translating this using his own symbols then it is easy to infer that we do
not have an example of an unprovable formula
Besides there is nothing that he writes with his symbols that cannot be translated into our usual
mathematical lingo and usual natural language symbols If he really had an example of such a
sentence why would he not present that to us even if at the end of his deductions in the simplest
way as possible
Without going into the merit of his writings for for instance the translator or even the assistant of
the translator could have swapped = 0 for something else in the remark of the item 28 we know that
what is being presented to us is not being presented to standards In Science communication is
obviously priority Many researchers have spent an entire life perfecting mathematical lingo for
712
instance so that everything would be as objective and universal as possible If something can be
passed onwards in a simpler form and we know that we have obligation of doing so if claiming to
be writing Science
Diagonalization that generates an unprovable sentence
From the writings that we had access to in the past not the original writings of Goumldel but writings
of other authors that wrote about his writings we understood that Goumldel simply used his codes to
make a list that were similar to the one presented by Cantor to prove that the real numbers
considered as a whole were uncountable
Cantor (see for instance [Eric Weisstein 1999]) would have tried to make a correspondence
between natural numbers and the real numbers and would have ended up proving that such was
impossible through forming a new number which was not part of his original list by picking one
digit from each position that corresponded to a diagonal line traced from the first number to the last
number he had put in the list and making his new number differ from that digit in that position
If his new number differs from each number in the list by the nth decimal digit where nth
corresponds to the ordinal number representing the listed number in his original list say then the
new number cannot be equal to any of the numbers previously listed by him
The work of diagonalization of Cantor which is what we have just mentioned seems unnecessary
to prove such an easy-to-understand claim What is being claimed is that we cannot assign ordinal
numbers (see [Eric Weisstein 1999b] for instance) in bijection or one-to-one correspondence to
any non-degenerated slice of the real numbers line and therefore we cannot count considering
how Mathematics has defined the operation of counting the real numbers not even in a small
interval like even in the smallest as possible interval that be non-degenerated
We could have proven that by simply listing numbers of the shape 0something as Cantor does but
812
with first decimal digit being the own natural number that we wish to use to count
It is easy to see that there are plenty of reals in between each two lines of the just-assembled list so
that the diagonal formation is unnecessary
In any hypothesis it is claimed that Goumldel used this diagonalization of Cantor to once more
exhibit an unprovable statement from the Arithmetic World
He would then have used his symbols and would have listed all possible formula of the type a + b =
c lets say
Obviously the case that the assignments of value to a b and c will use the entire spectrum of the
natural numbers so that varying only one of those variables in the formula already has covered the
ordinal numbers
Utilizing Cantors argument here means creating a formula that is not listed and claiming that we do
not have an ordinal for it
However Goumldel changes that into the formula not being passive of deduction from the previous
formulas in the list because it will differ from each one of those formulas in the list by the n th digit
where nth is the order in the list of each one of those formulas
Thinking about it leads us to doubt his statement straight away For instance the operation of
multiplication derives from the operation of summing We could have several lines of summing and
infer multiplication The only symbol in common would be the equal sign
2 x 12 = 24 comes from perhaps 2412 = 2
Considering a b and c here plus codes for the operations would lead us to having only the sign =
and the number 12 as common elements Yet one formula did come from the other
To eliminate the equal sign we can think of an example from Set Theory for instance
From 1 S a є S b є S 2 S c є S d є Snotin notin until 13 S y є S z є Snotin plus X = a b c
hellip z we infer X C S
Now if we consider the position of the elements in the line and symbols we will have no
912
intersection between those lines and the conclusion Yet we have no doubts that the conclusion does
derive from the premises
We have not seen the original works of Goumldel only had access to what other people have written
about his work on this proof but we understand that he either commits a mistake of the order that
we have just pointed or he forgets that the formula has to make sense for sometimes he is told to
have simply used all his symbols chaotically and produced a formula from a diagonal over the list at
the end
The main problem with this proof is the story of coding what is already coded basically
As said before his coding should only be considered scientifically acceptable if it could create a
formula that is unprovable inside of the metalanguage for Mathematics because we should always
go in the simplest way as possible in scientific argumentation and if his example involves what
could have been described with the standard codessymbols for Mathematics then it should have
been obviously
Conclusion
There is no actual scientific evidence on the incompleteness of axiomatic systems for Arithmetic
coming from the works of Goumldel therefore we must assume that the axiomatic system that we use
for Arithmetic is complete
1012
References
Eric Weisstein (1999) Cantor Diagonal Method Retrieved 14 November 2011
from MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCantorDiagonalMethodhtml
Eric Weisstein (1999b) Countable Set Retrieved 14 November 2011 from
MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCountableSethtml
Juliette Kennedy (2011) Kurt Goumldel The Stanford Encyclopedia of Philosophy (Fall 2011
Edition) Edward N Zalta (ed) Retrieved November 10 2011 from
httpplatostanfordeduarchivesfall2011entriesgoedel
Kurt Goumldel (1930) On Formally undecidable propositions of Principia Mathematica and related
systems 1 Yggdrasils WN Library Retrieved 10 November 2011 from
httpwwwgeierhuGOEDELGoumldel_origGoumldel3htm
Robert Constable (2009) Lecture 23 Unsolvable Problems in Logic Applied Logic course CS
4860 Department of Computer Science Cornell University Retrieved 14 November 2011 from
httpwwwcscornelleducoursescs48602009splec-23pdf
1112
S Feferman J D W Junior S C Kleene G H Moore R M Soloway and J V
Heijenoort (eds) (1986) Kurt Goumldel Collected Works volume I publications
1929-1936 Oxford University Press ISBN-13 978-0195039641
Postal address for the author
P O Box 12396 ABeckett st Melbourne Victoria Australia 3000
1212
does not belong to K therefore we have a proof of [R(q)q] what is absolutely consistent with our
assumption
5th line If on the contrary the negation of [R(q)q] were provable then n would not belong to K
ie we would not have a proof of [R(q)q]
Remarks If we can prove that the assumed ordering relation for q is not true then we obviously do
not have a proof of [R(q)q] and therefore according to the definition of the K class by the own
Goumldel in the discussed work q would belong to K (not n as it appears in the translated text but q
instead obviously) precisely the opposite to what is asserted (again)
6th line [R(q)q] would thus be provable at the same time as its negation which again is
impossible
Remarks Trivially there are absolutely no conflicts instead because it is not true that [R(q)q] and
its negation are provable at the same time
Goumldel never found an example of mathematical formula that were undecidable inside of Arithmetic
through the K-class example therefore
The self-referential sentence
In large amount of the popular texts (see for instance [Kleene et al 1986]) we find Goumldel being
mentioned as if he had created a proposition that went like this V = V is unprovable in Г
The argument presented when such a possibility is exhibited is that if V is evaluated as true then V
is unprovable what then is a proof of incompleteness On the other hand if V is evaluated as false
then V must be provable but it states that it is unprovable so that we are left with a proposition that
we cannot judge in terms of truth-value which is what we wanted to achieve in order to defend the
incompleteness of the axiomatic systems that refer to the natural numbers
The problems with the possible proof of incompleteness that we have just presented are several The
512
most obvious of them all is perhaps what we have already mentioned There is a temporal obstacle
that cannot be removed Basically we will have V containing no determined sentence in V is
unprovable in Г but V will become determined as we name the sentence that we have just
mentioned V what then makes the example be rejected by Science for we cannot assert something
scientifically about something that we have not yet defined
To mention one more argument we have the problem with the detachment issue we present in our
analysis of The Liar Scientific statements must be completely cold and an entity asserting things
about themselves is not what we could call cold Instead that would be the warmest situation of all
of unavoidable attachment as in opposition to detachment and we insist on detachment
impartiality in all senses being an absolutely necessary condition for us to claim to be doing
Science
The ω-consistency theorem consequence
Two basic problems prevent this argument from being sound The first problem is what we have
already mentioned We cannot really find any evidence that ω-consistency implies consistency The
second is that the symbols of Goumldel created apparently to codify formulas in Mathematics (good
question would be why like are they not already codes and best as possible) like to replace
mathematical symbols that are used world-wide such as ( with numbers seem to confound the own
Goumldel in worse ways than they confound us
It is trivially the case that anything that may be done with a new set of symbols for mathematical
formulas may be done with the current set of symbols so why is it that Goumldel would be worried
about creating a new set of symbols
Goumldel is apparently trying to refer to things like generalization of statements with his symbols For
instance what we have mentioned earlier on Suppose that α n + 2 and [α3] 3 + 2 We then know
612
that α generalizes [α3] Mathematics has to appeal to natural language in order to describe what is
going on there to perfection is it not Goumldel was not happy with that He wanted symbols to replace
those words (generalizes)
We then start thinking that his objective was finding a formula inside of the metalanguage that were
unprovable right
That is what everyone says (P This sentence is not provable)
However Goumldel actually always worked with his own symbols in this development and never left
the original symbols of Mathematics (nothing is outside of Mathematics when translated) reaching
the conclusion that 17 Gen r is not c-provable instead
From this supposed finding he generalizes to v Gen r not being decidable
When we go through his symbols and try to work out what this so few symbols together mean the
disappointment cannot be avoided
We go translating his lingo using his list of symbols and reach a term that cannot be translated by
step six of our translation processes which is n St 17 = 0 (we are presented with item 28 k St v x
but we see no sign of anything like k St v)
If we cannot find a way of translating this using his own symbols then it is easy to infer that we do
not have an example of an unprovable formula
Besides there is nothing that he writes with his symbols that cannot be translated into our usual
mathematical lingo and usual natural language symbols If he really had an example of such a
sentence why would he not present that to us even if at the end of his deductions in the simplest
way as possible
Without going into the merit of his writings for for instance the translator or even the assistant of
the translator could have swapped = 0 for something else in the remark of the item 28 we know that
what is being presented to us is not being presented to standards In Science communication is
obviously priority Many researchers have spent an entire life perfecting mathematical lingo for
712
instance so that everything would be as objective and universal as possible If something can be
passed onwards in a simpler form and we know that we have obligation of doing so if claiming to
be writing Science
Diagonalization that generates an unprovable sentence
From the writings that we had access to in the past not the original writings of Goumldel but writings
of other authors that wrote about his writings we understood that Goumldel simply used his codes to
make a list that were similar to the one presented by Cantor to prove that the real numbers
considered as a whole were uncountable
Cantor (see for instance [Eric Weisstein 1999]) would have tried to make a correspondence
between natural numbers and the real numbers and would have ended up proving that such was
impossible through forming a new number which was not part of his original list by picking one
digit from each position that corresponded to a diagonal line traced from the first number to the last
number he had put in the list and making his new number differ from that digit in that position
If his new number differs from each number in the list by the nth decimal digit where nth
corresponds to the ordinal number representing the listed number in his original list say then the
new number cannot be equal to any of the numbers previously listed by him
The work of diagonalization of Cantor which is what we have just mentioned seems unnecessary
to prove such an easy-to-understand claim What is being claimed is that we cannot assign ordinal
numbers (see [Eric Weisstein 1999b] for instance) in bijection or one-to-one correspondence to
any non-degenerated slice of the real numbers line and therefore we cannot count considering
how Mathematics has defined the operation of counting the real numbers not even in a small
interval like even in the smallest as possible interval that be non-degenerated
We could have proven that by simply listing numbers of the shape 0something as Cantor does but
812
with first decimal digit being the own natural number that we wish to use to count
It is easy to see that there are plenty of reals in between each two lines of the just-assembled list so
that the diagonal formation is unnecessary
In any hypothesis it is claimed that Goumldel used this diagonalization of Cantor to once more
exhibit an unprovable statement from the Arithmetic World
He would then have used his symbols and would have listed all possible formula of the type a + b =
c lets say
Obviously the case that the assignments of value to a b and c will use the entire spectrum of the
natural numbers so that varying only one of those variables in the formula already has covered the
ordinal numbers
Utilizing Cantors argument here means creating a formula that is not listed and claiming that we do
not have an ordinal for it
However Goumldel changes that into the formula not being passive of deduction from the previous
formulas in the list because it will differ from each one of those formulas in the list by the n th digit
where nth is the order in the list of each one of those formulas
Thinking about it leads us to doubt his statement straight away For instance the operation of
multiplication derives from the operation of summing We could have several lines of summing and
infer multiplication The only symbol in common would be the equal sign
2 x 12 = 24 comes from perhaps 2412 = 2
Considering a b and c here plus codes for the operations would lead us to having only the sign =
and the number 12 as common elements Yet one formula did come from the other
To eliminate the equal sign we can think of an example from Set Theory for instance
From 1 S a є S b є S 2 S c є S d є Snotin notin until 13 S y є S z є Snotin plus X = a b c
hellip z we infer X C S
Now if we consider the position of the elements in the line and symbols we will have no
912
intersection between those lines and the conclusion Yet we have no doubts that the conclusion does
derive from the premises
We have not seen the original works of Goumldel only had access to what other people have written
about his work on this proof but we understand that he either commits a mistake of the order that
we have just pointed or he forgets that the formula has to make sense for sometimes he is told to
have simply used all his symbols chaotically and produced a formula from a diagonal over the list at
the end
The main problem with this proof is the story of coding what is already coded basically
As said before his coding should only be considered scientifically acceptable if it could create a
formula that is unprovable inside of the metalanguage for Mathematics because we should always
go in the simplest way as possible in scientific argumentation and if his example involves what
could have been described with the standard codessymbols for Mathematics then it should have
been obviously
Conclusion
There is no actual scientific evidence on the incompleteness of axiomatic systems for Arithmetic
coming from the works of Goumldel therefore we must assume that the axiomatic system that we use
for Arithmetic is complete
1012
References
Eric Weisstein (1999) Cantor Diagonal Method Retrieved 14 November 2011
from MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCantorDiagonalMethodhtml
Eric Weisstein (1999b) Countable Set Retrieved 14 November 2011 from
MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCountableSethtml
Juliette Kennedy (2011) Kurt Goumldel The Stanford Encyclopedia of Philosophy (Fall 2011
Edition) Edward N Zalta (ed) Retrieved November 10 2011 from
httpplatostanfordeduarchivesfall2011entriesgoedel
Kurt Goumldel (1930) On Formally undecidable propositions of Principia Mathematica and related
systems 1 Yggdrasils WN Library Retrieved 10 November 2011 from
httpwwwgeierhuGOEDELGoumldel_origGoumldel3htm
Robert Constable (2009) Lecture 23 Unsolvable Problems in Logic Applied Logic course CS
4860 Department of Computer Science Cornell University Retrieved 14 November 2011 from
httpwwwcscornelleducoursescs48602009splec-23pdf
1112
S Feferman J D W Junior S C Kleene G H Moore R M Soloway and J V
Heijenoort (eds) (1986) Kurt Goumldel Collected Works volume I publications
1929-1936 Oxford University Press ISBN-13 978-0195039641
Postal address for the author
P O Box 12396 ABeckett st Melbourne Victoria Australia 3000
1212
most obvious of them all is perhaps what we have already mentioned There is a temporal obstacle
that cannot be removed Basically we will have V containing no determined sentence in V is
unprovable in Г but V will become determined as we name the sentence that we have just
mentioned V what then makes the example be rejected by Science for we cannot assert something
scientifically about something that we have not yet defined
To mention one more argument we have the problem with the detachment issue we present in our
analysis of The Liar Scientific statements must be completely cold and an entity asserting things
about themselves is not what we could call cold Instead that would be the warmest situation of all
of unavoidable attachment as in opposition to detachment and we insist on detachment
impartiality in all senses being an absolutely necessary condition for us to claim to be doing
Science
The ω-consistency theorem consequence
Two basic problems prevent this argument from being sound The first problem is what we have
already mentioned We cannot really find any evidence that ω-consistency implies consistency The
second is that the symbols of Goumldel created apparently to codify formulas in Mathematics (good
question would be why like are they not already codes and best as possible) like to replace
mathematical symbols that are used world-wide such as ( with numbers seem to confound the own
Goumldel in worse ways than they confound us
It is trivially the case that anything that may be done with a new set of symbols for mathematical
formulas may be done with the current set of symbols so why is it that Goumldel would be worried
about creating a new set of symbols
Goumldel is apparently trying to refer to things like generalization of statements with his symbols For
instance what we have mentioned earlier on Suppose that α n + 2 and [α3] 3 + 2 We then know
612
that α generalizes [α3] Mathematics has to appeal to natural language in order to describe what is
going on there to perfection is it not Goumldel was not happy with that He wanted symbols to replace
those words (generalizes)
We then start thinking that his objective was finding a formula inside of the metalanguage that were
unprovable right
That is what everyone says (P This sentence is not provable)
However Goumldel actually always worked with his own symbols in this development and never left
the original symbols of Mathematics (nothing is outside of Mathematics when translated) reaching
the conclusion that 17 Gen r is not c-provable instead
From this supposed finding he generalizes to v Gen r not being decidable
When we go through his symbols and try to work out what this so few symbols together mean the
disappointment cannot be avoided
We go translating his lingo using his list of symbols and reach a term that cannot be translated by
step six of our translation processes which is n St 17 = 0 (we are presented with item 28 k St v x
but we see no sign of anything like k St v)
If we cannot find a way of translating this using his own symbols then it is easy to infer that we do
not have an example of an unprovable formula
Besides there is nothing that he writes with his symbols that cannot be translated into our usual
mathematical lingo and usual natural language symbols If he really had an example of such a
sentence why would he not present that to us even if at the end of his deductions in the simplest
way as possible
Without going into the merit of his writings for for instance the translator or even the assistant of
the translator could have swapped = 0 for something else in the remark of the item 28 we know that
what is being presented to us is not being presented to standards In Science communication is
obviously priority Many researchers have spent an entire life perfecting mathematical lingo for
712
instance so that everything would be as objective and universal as possible If something can be
passed onwards in a simpler form and we know that we have obligation of doing so if claiming to
be writing Science
Diagonalization that generates an unprovable sentence
From the writings that we had access to in the past not the original writings of Goumldel but writings
of other authors that wrote about his writings we understood that Goumldel simply used his codes to
make a list that were similar to the one presented by Cantor to prove that the real numbers
considered as a whole were uncountable
Cantor (see for instance [Eric Weisstein 1999]) would have tried to make a correspondence
between natural numbers and the real numbers and would have ended up proving that such was
impossible through forming a new number which was not part of his original list by picking one
digit from each position that corresponded to a diagonal line traced from the first number to the last
number he had put in the list and making his new number differ from that digit in that position
If his new number differs from each number in the list by the nth decimal digit where nth
corresponds to the ordinal number representing the listed number in his original list say then the
new number cannot be equal to any of the numbers previously listed by him
The work of diagonalization of Cantor which is what we have just mentioned seems unnecessary
to prove such an easy-to-understand claim What is being claimed is that we cannot assign ordinal
numbers (see [Eric Weisstein 1999b] for instance) in bijection or one-to-one correspondence to
any non-degenerated slice of the real numbers line and therefore we cannot count considering
how Mathematics has defined the operation of counting the real numbers not even in a small
interval like even in the smallest as possible interval that be non-degenerated
We could have proven that by simply listing numbers of the shape 0something as Cantor does but
812
with first decimal digit being the own natural number that we wish to use to count
It is easy to see that there are plenty of reals in between each two lines of the just-assembled list so
that the diagonal formation is unnecessary
In any hypothesis it is claimed that Goumldel used this diagonalization of Cantor to once more
exhibit an unprovable statement from the Arithmetic World
He would then have used his symbols and would have listed all possible formula of the type a + b =
c lets say
Obviously the case that the assignments of value to a b and c will use the entire spectrum of the
natural numbers so that varying only one of those variables in the formula already has covered the
ordinal numbers
Utilizing Cantors argument here means creating a formula that is not listed and claiming that we do
not have an ordinal for it
However Goumldel changes that into the formula not being passive of deduction from the previous
formulas in the list because it will differ from each one of those formulas in the list by the n th digit
where nth is the order in the list of each one of those formulas
Thinking about it leads us to doubt his statement straight away For instance the operation of
multiplication derives from the operation of summing We could have several lines of summing and
infer multiplication The only symbol in common would be the equal sign
2 x 12 = 24 comes from perhaps 2412 = 2
Considering a b and c here plus codes for the operations would lead us to having only the sign =
and the number 12 as common elements Yet one formula did come from the other
To eliminate the equal sign we can think of an example from Set Theory for instance
From 1 S a є S b є S 2 S c є S d є Snotin notin until 13 S y є S z є Snotin plus X = a b c
hellip z we infer X C S
Now if we consider the position of the elements in the line and symbols we will have no
912
intersection between those lines and the conclusion Yet we have no doubts that the conclusion does
derive from the premises
We have not seen the original works of Goumldel only had access to what other people have written
about his work on this proof but we understand that he either commits a mistake of the order that
we have just pointed or he forgets that the formula has to make sense for sometimes he is told to
have simply used all his symbols chaotically and produced a formula from a diagonal over the list at
the end
The main problem with this proof is the story of coding what is already coded basically
As said before his coding should only be considered scientifically acceptable if it could create a
formula that is unprovable inside of the metalanguage for Mathematics because we should always
go in the simplest way as possible in scientific argumentation and if his example involves what
could have been described with the standard codessymbols for Mathematics then it should have
been obviously
Conclusion
There is no actual scientific evidence on the incompleteness of axiomatic systems for Arithmetic
coming from the works of Goumldel therefore we must assume that the axiomatic system that we use
for Arithmetic is complete
1012
References
Eric Weisstein (1999) Cantor Diagonal Method Retrieved 14 November 2011
from MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCantorDiagonalMethodhtml
Eric Weisstein (1999b) Countable Set Retrieved 14 November 2011 from
MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCountableSethtml
Juliette Kennedy (2011) Kurt Goumldel The Stanford Encyclopedia of Philosophy (Fall 2011
Edition) Edward N Zalta (ed) Retrieved November 10 2011 from
httpplatostanfordeduarchivesfall2011entriesgoedel
Kurt Goumldel (1930) On Formally undecidable propositions of Principia Mathematica and related
systems 1 Yggdrasils WN Library Retrieved 10 November 2011 from
httpwwwgeierhuGOEDELGoumldel_origGoumldel3htm
Robert Constable (2009) Lecture 23 Unsolvable Problems in Logic Applied Logic course CS
4860 Department of Computer Science Cornell University Retrieved 14 November 2011 from
httpwwwcscornelleducoursescs48602009splec-23pdf
1112
S Feferman J D W Junior S C Kleene G H Moore R M Soloway and J V
Heijenoort (eds) (1986) Kurt Goumldel Collected Works volume I publications
1929-1936 Oxford University Press ISBN-13 978-0195039641
Postal address for the author
P O Box 12396 ABeckett st Melbourne Victoria Australia 3000
1212
that α generalizes [α3] Mathematics has to appeal to natural language in order to describe what is
going on there to perfection is it not Goumldel was not happy with that He wanted symbols to replace
those words (generalizes)
We then start thinking that his objective was finding a formula inside of the metalanguage that were
unprovable right
That is what everyone says (P This sentence is not provable)
However Goumldel actually always worked with his own symbols in this development and never left
the original symbols of Mathematics (nothing is outside of Mathematics when translated) reaching
the conclusion that 17 Gen r is not c-provable instead
From this supposed finding he generalizes to v Gen r not being decidable
When we go through his symbols and try to work out what this so few symbols together mean the
disappointment cannot be avoided
We go translating his lingo using his list of symbols and reach a term that cannot be translated by
step six of our translation processes which is n St 17 = 0 (we are presented with item 28 k St v x
but we see no sign of anything like k St v)
If we cannot find a way of translating this using his own symbols then it is easy to infer that we do
not have an example of an unprovable formula
Besides there is nothing that he writes with his symbols that cannot be translated into our usual
mathematical lingo and usual natural language symbols If he really had an example of such a
sentence why would he not present that to us even if at the end of his deductions in the simplest
way as possible
Without going into the merit of his writings for for instance the translator or even the assistant of
the translator could have swapped = 0 for something else in the remark of the item 28 we know that
what is being presented to us is not being presented to standards In Science communication is
obviously priority Many researchers have spent an entire life perfecting mathematical lingo for
712
instance so that everything would be as objective and universal as possible If something can be
passed onwards in a simpler form and we know that we have obligation of doing so if claiming to
be writing Science
Diagonalization that generates an unprovable sentence
From the writings that we had access to in the past not the original writings of Goumldel but writings
of other authors that wrote about his writings we understood that Goumldel simply used his codes to
make a list that were similar to the one presented by Cantor to prove that the real numbers
considered as a whole were uncountable
Cantor (see for instance [Eric Weisstein 1999]) would have tried to make a correspondence
between natural numbers and the real numbers and would have ended up proving that such was
impossible through forming a new number which was not part of his original list by picking one
digit from each position that corresponded to a diagonal line traced from the first number to the last
number he had put in the list and making his new number differ from that digit in that position
If his new number differs from each number in the list by the nth decimal digit where nth
corresponds to the ordinal number representing the listed number in his original list say then the
new number cannot be equal to any of the numbers previously listed by him
The work of diagonalization of Cantor which is what we have just mentioned seems unnecessary
to prove such an easy-to-understand claim What is being claimed is that we cannot assign ordinal
numbers (see [Eric Weisstein 1999b] for instance) in bijection or one-to-one correspondence to
any non-degenerated slice of the real numbers line and therefore we cannot count considering
how Mathematics has defined the operation of counting the real numbers not even in a small
interval like even in the smallest as possible interval that be non-degenerated
We could have proven that by simply listing numbers of the shape 0something as Cantor does but
812
with first decimal digit being the own natural number that we wish to use to count
It is easy to see that there are plenty of reals in between each two lines of the just-assembled list so
that the diagonal formation is unnecessary
In any hypothesis it is claimed that Goumldel used this diagonalization of Cantor to once more
exhibit an unprovable statement from the Arithmetic World
He would then have used his symbols and would have listed all possible formula of the type a + b =
c lets say
Obviously the case that the assignments of value to a b and c will use the entire spectrum of the
natural numbers so that varying only one of those variables in the formula already has covered the
ordinal numbers
Utilizing Cantors argument here means creating a formula that is not listed and claiming that we do
not have an ordinal for it
However Goumldel changes that into the formula not being passive of deduction from the previous
formulas in the list because it will differ from each one of those formulas in the list by the n th digit
where nth is the order in the list of each one of those formulas
Thinking about it leads us to doubt his statement straight away For instance the operation of
multiplication derives from the operation of summing We could have several lines of summing and
infer multiplication The only symbol in common would be the equal sign
2 x 12 = 24 comes from perhaps 2412 = 2
Considering a b and c here plus codes for the operations would lead us to having only the sign =
and the number 12 as common elements Yet one formula did come from the other
To eliminate the equal sign we can think of an example from Set Theory for instance
From 1 S a є S b є S 2 S c є S d є Snotin notin until 13 S y є S z є Snotin plus X = a b c
hellip z we infer X C S
Now if we consider the position of the elements in the line and symbols we will have no
912
intersection between those lines and the conclusion Yet we have no doubts that the conclusion does
derive from the premises
We have not seen the original works of Goumldel only had access to what other people have written
about his work on this proof but we understand that he either commits a mistake of the order that
we have just pointed or he forgets that the formula has to make sense for sometimes he is told to
have simply used all his symbols chaotically and produced a formula from a diagonal over the list at
the end
The main problem with this proof is the story of coding what is already coded basically
As said before his coding should only be considered scientifically acceptable if it could create a
formula that is unprovable inside of the metalanguage for Mathematics because we should always
go in the simplest way as possible in scientific argumentation and if his example involves what
could have been described with the standard codessymbols for Mathematics then it should have
been obviously
Conclusion
There is no actual scientific evidence on the incompleteness of axiomatic systems for Arithmetic
coming from the works of Goumldel therefore we must assume that the axiomatic system that we use
for Arithmetic is complete
1012
References
Eric Weisstein (1999) Cantor Diagonal Method Retrieved 14 November 2011
from MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCantorDiagonalMethodhtml
Eric Weisstein (1999b) Countable Set Retrieved 14 November 2011 from
MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCountableSethtml
Juliette Kennedy (2011) Kurt Goumldel The Stanford Encyclopedia of Philosophy (Fall 2011
Edition) Edward N Zalta (ed) Retrieved November 10 2011 from
httpplatostanfordeduarchivesfall2011entriesgoedel
Kurt Goumldel (1930) On Formally undecidable propositions of Principia Mathematica and related
systems 1 Yggdrasils WN Library Retrieved 10 November 2011 from
httpwwwgeierhuGOEDELGoumldel_origGoumldel3htm
Robert Constable (2009) Lecture 23 Unsolvable Problems in Logic Applied Logic course CS
4860 Department of Computer Science Cornell University Retrieved 14 November 2011 from
httpwwwcscornelleducoursescs48602009splec-23pdf
1112
S Feferman J D W Junior S C Kleene G H Moore R M Soloway and J V
Heijenoort (eds) (1986) Kurt Goumldel Collected Works volume I publications
1929-1936 Oxford University Press ISBN-13 978-0195039641
Postal address for the author
P O Box 12396 ABeckett st Melbourne Victoria Australia 3000
1212
instance so that everything would be as objective and universal as possible If something can be
passed onwards in a simpler form and we know that we have obligation of doing so if claiming to
be writing Science
Diagonalization that generates an unprovable sentence
From the writings that we had access to in the past not the original writings of Goumldel but writings
of other authors that wrote about his writings we understood that Goumldel simply used his codes to
make a list that were similar to the one presented by Cantor to prove that the real numbers
considered as a whole were uncountable
Cantor (see for instance [Eric Weisstein 1999]) would have tried to make a correspondence
between natural numbers and the real numbers and would have ended up proving that such was
impossible through forming a new number which was not part of his original list by picking one
digit from each position that corresponded to a diagonal line traced from the first number to the last
number he had put in the list and making his new number differ from that digit in that position
If his new number differs from each number in the list by the nth decimal digit where nth
corresponds to the ordinal number representing the listed number in his original list say then the
new number cannot be equal to any of the numbers previously listed by him
The work of diagonalization of Cantor which is what we have just mentioned seems unnecessary
to prove such an easy-to-understand claim What is being claimed is that we cannot assign ordinal
numbers (see [Eric Weisstein 1999b] for instance) in bijection or one-to-one correspondence to
any non-degenerated slice of the real numbers line and therefore we cannot count considering
how Mathematics has defined the operation of counting the real numbers not even in a small
interval like even in the smallest as possible interval that be non-degenerated
We could have proven that by simply listing numbers of the shape 0something as Cantor does but
812
with first decimal digit being the own natural number that we wish to use to count
It is easy to see that there are plenty of reals in between each two lines of the just-assembled list so
that the diagonal formation is unnecessary
In any hypothesis it is claimed that Goumldel used this diagonalization of Cantor to once more
exhibit an unprovable statement from the Arithmetic World
He would then have used his symbols and would have listed all possible formula of the type a + b =
c lets say
Obviously the case that the assignments of value to a b and c will use the entire spectrum of the
natural numbers so that varying only one of those variables in the formula already has covered the
ordinal numbers
Utilizing Cantors argument here means creating a formula that is not listed and claiming that we do
not have an ordinal for it
However Goumldel changes that into the formula not being passive of deduction from the previous
formulas in the list because it will differ from each one of those formulas in the list by the n th digit
where nth is the order in the list of each one of those formulas
Thinking about it leads us to doubt his statement straight away For instance the operation of
multiplication derives from the operation of summing We could have several lines of summing and
infer multiplication The only symbol in common would be the equal sign
2 x 12 = 24 comes from perhaps 2412 = 2
Considering a b and c here plus codes for the operations would lead us to having only the sign =
and the number 12 as common elements Yet one formula did come from the other
To eliminate the equal sign we can think of an example from Set Theory for instance
From 1 S a є S b є S 2 S c є S d є Snotin notin until 13 S y є S z є Snotin plus X = a b c
hellip z we infer X C S
Now if we consider the position of the elements in the line and symbols we will have no
912
intersection between those lines and the conclusion Yet we have no doubts that the conclusion does
derive from the premises
We have not seen the original works of Goumldel only had access to what other people have written
about his work on this proof but we understand that he either commits a mistake of the order that
we have just pointed or he forgets that the formula has to make sense for sometimes he is told to
have simply used all his symbols chaotically and produced a formula from a diagonal over the list at
the end
The main problem with this proof is the story of coding what is already coded basically
As said before his coding should only be considered scientifically acceptable if it could create a
formula that is unprovable inside of the metalanguage for Mathematics because we should always
go in the simplest way as possible in scientific argumentation and if his example involves what
could have been described with the standard codessymbols for Mathematics then it should have
been obviously
Conclusion
There is no actual scientific evidence on the incompleteness of axiomatic systems for Arithmetic
coming from the works of Goumldel therefore we must assume that the axiomatic system that we use
for Arithmetic is complete
1012
References
Eric Weisstein (1999) Cantor Diagonal Method Retrieved 14 November 2011
from MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCantorDiagonalMethodhtml
Eric Weisstein (1999b) Countable Set Retrieved 14 November 2011 from
MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCountableSethtml
Juliette Kennedy (2011) Kurt Goumldel The Stanford Encyclopedia of Philosophy (Fall 2011
Edition) Edward N Zalta (ed) Retrieved November 10 2011 from
httpplatostanfordeduarchivesfall2011entriesgoedel
Kurt Goumldel (1930) On Formally undecidable propositions of Principia Mathematica and related
systems 1 Yggdrasils WN Library Retrieved 10 November 2011 from
httpwwwgeierhuGOEDELGoumldel_origGoumldel3htm
Robert Constable (2009) Lecture 23 Unsolvable Problems in Logic Applied Logic course CS
4860 Department of Computer Science Cornell University Retrieved 14 November 2011 from
httpwwwcscornelleducoursescs48602009splec-23pdf
1112
S Feferman J D W Junior S C Kleene G H Moore R M Soloway and J V
Heijenoort (eds) (1986) Kurt Goumldel Collected Works volume I publications
1929-1936 Oxford University Press ISBN-13 978-0195039641
Postal address for the author
P O Box 12396 ABeckett st Melbourne Victoria Australia 3000
1212
with first decimal digit being the own natural number that we wish to use to count
It is easy to see that there are plenty of reals in between each two lines of the just-assembled list so
that the diagonal formation is unnecessary
In any hypothesis it is claimed that Goumldel used this diagonalization of Cantor to once more
exhibit an unprovable statement from the Arithmetic World
He would then have used his symbols and would have listed all possible formula of the type a + b =
c lets say
Obviously the case that the assignments of value to a b and c will use the entire spectrum of the
natural numbers so that varying only one of those variables in the formula already has covered the
ordinal numbers
Utilizing Cantors argument here means creating a formula that is not listed and claiming that we do
not have an ordinal for it
However Goumldel changes that into the formula not being passive of deduction from the previous
formulas in the list because it will differ from each one of those formulas in the list by the n th digit
where nth is the order in the list of each one of those formulas
Thinking about it leads us to doubt his statement straight away For instance the operation of
multiplication derives from the operation of summing We could have several lines of summing and
infer multiplication The only symbol in common would be the equal sign
2 x 12 = 24 comes from perhaps 2412 = 2
Considering a b and c here plus codes for the operations would lead us to having only the sign =
and the number 12 as common elements Yet one formula did come from the other
To eliminate the equal sign we can think of an example from Set Theory for instance
From 1 S a є S b є S 2 S c є S d є Snotin notin until 13 S y є S z є Snotin plus X = a b c
hellip z we infer X C S
Now if we consider the position of the elements in the line and symbols we will have no
912
intersection between those lines and the conclusion Yet we have no doubts that the conclusion does
derive from the premises
We have not seen the original works of Goumldel only had access to what other people have written
about his work on this proof but we understand that he either commits a mistake of the order that
we have just pointed or he forgets that the formula has to make sense for sometimes he is told to
have simply used all his symbols chaotically and produced a formula from a diagonal over the list at
the end
The main problem with this proof is the story of coding what is already coded basically
As said before his coding should only be considered scientifically acceptable if it could create a
formula that is unprovable inside of the metalanguage for Mathematics because we should always
go in the simplest way as possible in scientific argumentation and if his example involves what
could have been described with the standard codessymbols for Mathematics then it should have
been obviously
Conclusion
There is no actual scientific evidence on the incompleteness of axiomatic systems for Arithmetic
coming from the works of Goumldel therefore we must assume that the axiomatic system that we use
for Arithmetic is complete
1012
References
Eric Weisstein (1999) Cantor Diagonal Method Retrieved 14 November 2011
from MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCantorDiagonalMethodhtml
Eric Weisstein (1999b) Countable Set Retrieved 14 November 2011 from
MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCountableSethtml
Juliette Kennedy (2011) Kurt Goumldel The Stanford Encyclopedia of Philosophy (Fall 2011
Edition) Edward N Zalta (ed) Retrieved November 10 2011 from
httpplatostanfordeduarchivesfall2011entriesgoedel
Kurt Goumldel (1930) On Formally undecidable propositions of Principia Mathematica and related
systems 1 Yggdrasils WN Library Retrieved 10 November 2011 from
httpwwwgeierhuGOEDELGoumldel_origGoumldel3htm
Robert Constable (2009) Lecture 23 Unsolvable Problems in Logic Applied Logic course CS
4860 Department of Computer Science Cornell University Retrieved 14 November 2011 from
httpwwwcscornelleducoursescs48602009splec-23pdf
1112
S Feferman J D W Junior S C Kleene G H Moore R M Soloway and J V
Heijenoort (eds) (1986) Kurt Goumldel Collected Works volume I publications
1929-1936 Oxford University Press ISBN-13 978-0195039641
Postal address for the author
P O Box 12396 ABeckett st Melbourne Victoria Australia 3000
1212
intersection between those lines and the conclusion Yet we have no doubts that the conclusion does
derive from the premises
We have not seen the original works of Goumldel only had access to what other people have written
about his work on this proof but we understand that he either commits a mistake of the order that
we have just pointed or he forgets that the formula has to make sense for sometimes he is told to
have simply used all his symbols chaotically and produced a formula from a diagonal over the list at
the end
The main problem with this proof is the story of coding what is already coded basically
As said before his coding should only be considered scientifically acceptable if it could create a
formula that is unprovable inside of the metalanguage for Mathematics because we should always
go in the simplest way as possible in scientific argumentation and if his example involves what
could have been described with the standard codessymbols for Mathematics then it should have
been obviously
Conclusion
There is no actual scientific evidence on the incompleteness of axiomatic systems for Arithmetic
coming from the works of Goumldel therefore we must assume that the axiomatic system that we use
for Arithmetic is complete
1012
References
Eric Weisstein (1999) Cantor Diagonal Method Retrieved 14 November 2011
from MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCantorDiagonalMethodhtml
Eric Weisstein (1999b) Countable Set Retrieved 14 November 2011 from
MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCountableSethtml
Juliette Kennedy (2011) Kurt Goumldel The Stanford Encyclopedia of Philosophy (Fall 2011
Edition) Edward N Zalta (ed) Retrieved November 10 2011 from
httpplatostanfordeduarchivesfall2011entriesgoedel
Kurt Goumldel (1930) On Formally undecidable propositions of Principia Mathematica and related
systems 1 Yggdrasils WN Library Retrieved 10 November 2011 from
httpwwwgeierhuGOEDELGoumldel_origGoumldel3htm
Robert Constable (2009) Lecture 23 Unsolvable Problems in Logic Applied Logic course CS
4860 Department of Computer Science Cornell University Retrieved 14 November 2011 from
httpwwwcscornelleducoursescs48602009splec-23pdf
1112
S Feferman J D W Junior S C Kleene G H Moore R M Soloway and J V
Heijenoort (eds) (1986) Kurt Goumldel Collected Works volume I publications
1929-1936 Oxford University Press ISBN-13 978-0195039641
Postal address for the author
P O Box 12396 ABeckett st Melbourne Victoria Australia 3000
1212
References
Eric Weisstein (1999) Cantor Diagonal Method Retrieved 14 November 2011
from MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCantorDiagonalMethodhtml
Eric Weisstein (1999b) Countable Set Retrieved 14 November 2011 from
MathWorld--A Wolfram Web Resource
httpmathworldwolframcomCountableSethtml
Juliette Kennedy (2011) Kurt Goumldel The Stanford Encyclopedia of Philosophy (Fall 2011
Edition) Edward N Zalta (ed) Retrieved November 10 2011 from
httpplatostanfordeduarchivesfall2011entriesgoedel
Kurt Goumldel (1930) On Formally undecidable propositions of Principia Mathematica and related
systems 1 Yggdrasils WN Library Retrieved 10 November 2011 from
httpwwwgeierhuGOEDELGoumldel_origGoumldel3htm
Robert Constable (2009) Lecture 23 Unsolvable Problems in Logic Applied Logic course CS
4860 Department of Computer Science Cornell University Retrieved 14 November 2011 from
httpwwwcscornelleducoursescs48602009splec-23pdf
1112
S Feferman J D W Junior S C Kleene G H Moore R M Soloway and J V
Heijenoort (eds) (1986) Kurt Goumldel Collected Works volume I publications
1929-1936 Oxford University Press ISBN-13 978-0195039641
Postal address for the author
P O Box 12396 ABeckett st Melbourne Victoria Australia 3000
1212
S Feferman J D W Junior S C Kleene G H Moore R M Soloway and J V
Heijenoort (eds) (1986) Kurt Goumldel Collected Works volume I publications
1929-1936 Oxford University Press ISBN-13 978-0195039641
Postal address for the author
P O Box 12396 ABeckett st Melbourne Victoria Australia 3000
