Class 34:ProvingUnprovability

cs1120 Fall 2011David Evans11 - 11 - 11

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Plan

Explanation of PS8 OptionsUnprovability

Any questions about the interpreter?

Comments about Ivan’s lecture Wednesday

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Problem Set 8

Option J:Aazda (Charme)

Interpreter in Java + static type

checking

Option C:Conveying Computing

Option W:Web

Application

cs2110: Software Development Methods

Only Option J automatically satisfies the prerequisite for taking cs2110 this Spring. If you indicated interest in Computer Science major on your PS0 survey you are expected to do Option J. If you prefer to do a different option, must provide a convincing reason why.

Meta-Circularity?

Much of the course so far:Getting comfortable with recursive definitionsLearning to write programs to do (almost) anything

Starting today and next week:Getting un-comfortable with recursive definitionsThings no program can do!

Computer Science/Mathematics

Computer Science (Imperative Knowledge)Are there (well-defined) problems that

cannot be solved by any procedure?

Mathematics (Declarative Knowledge)Are there true conjectures that cannot be the

shown using any proof?Toda

yM

onda

y

Mechanical ReasoningAristotle (~350BC): Organon

Codify logical deduction with rules of inference (syllogisms)

Every A is a P

X is an A

X is a P

Premises

Conclusion

Every human is mortal. Gödel is human.Gödel is mortal.

Euclid (~300BC): ElementsReduce geometry to a few

axioms and derive the rest by following rules

Newton (1687): Philosophiæ Naturalis Principia Mathematica

Reduce the motion of objects (including planets) to following

axioms (laws) mechanically

Mechanical Reasoning

1800s – mathematicians work on codifying “laws of reasoning”

Augustus De Morgan (1806-1871)De Morgan’s lawsproof by induction

George Boole (1815-1864)Laws of Thought

Bertrand Russell (1872-1970)

1910-1913: Principia Mathematica (with Alfred Whitehead)

1918: Imprisoned for pacifism1950: Nobel Prize in Literature1955: Russell-Einstein Manifesto 1967: War Crimes in Vietnam

Note: this is the same Russell who wrote In Praise of Idleness!

When Einstein said, “Great spirits have always encountered violent opposition from mediocre minds.” he was talking about Bertrand Russell.

All true statements about numbers

Perfect Axiomatic System

Derives all true statements, and no false

statements starting from a finite number of axioms

and following mechanical inference rules.

Incomplete Axiomatic System

Derives some, but not all true

statements, and no false statements starting from a

finite number of axioms and following mechanical

inference rules.

incomplete

Inconsistent Axiomatic System

Derives all true statements, and some false statements starting from a

finite number of axioms and following mechanical

inference rules.some false statements

Principia Mathematica [1910]

Bertrand Russell(1872-1970)

Alfred Whitehead(1861-1947)

2000 pages

Attempted to axiomatize mathematical reasoning

Claimed to be complete and consistent:All true theorems could be derivedNo falsehoods could be derived

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Proving 1+1 = 2

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More Understandable Proof

Define the natural numbers

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More Understandable Proof

Define the natural numbers

Peano’ s Postulates:

N is the smallest set satisfying these postulates: P1. 1 is in N . P2. If x is in N , then its "successor" (succ x) is in N . P3. There is no x such that (succ x) = 1. P4. If x is not 1, then there is a y in N such that (succ y) = x. P5. If S is a subset of N , 1 is in S, and the implication

(x in S=> (succ x) in S) holds, then S=N.

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Proving 1+1 = 2

Define +: N × N N N is the smallest set satisfying these postulates: 1. 1 is in N . 2. If x is in N , then its "successor"

(succ x) is in N . 3. There is no x such that (succ x) =

1. 4. If x is not 1, then there is a y in

N such that (succ y) = x. 5. If S is a subset of N , 1 is in S,

and the implication (x in S=> (succ x) in S) holds, then S=N.

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Proving 1+1 = 2

Define +: N × N N

Call the inputs a and b.

If b is equal to 1: (+ a b) = (succ a)Otherwise: by P4, there exists c such that b = (succ c) (+ a b) = (succ (+ a c))

N is the smallest set satisfying these postulates: 1. 1 is in N . 2. If x is in N , then its "successor"

(succ x) is in N . 3. There is no x such that (succ x) =

1. 4. If x is not 1, then there is a y in

N such that (succ y) = x. 5. If S is a subset of N , 1 is in S,

and the implication (x in S=> (succ x) in S) holds, then S=N.

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Now the Proof!

Definition of (+ a b):If b is equal to 1: (+ a b) = (succ a)Otherwise: by P4, there exists c such that b = (succ c) (+ a b) = (succ (+ a c))

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Now the Proof!

“2” = (succ 1)1 + 1 = (succ 1)By definition of +,

1 + 1 = (succ 1) = “2”QED!

Definition of (+ a b):If b is equal to 1: (+ a b) = (succ a)Otherwise: by P4, there exists c such that b = (succ c) (+ a b) = (succ (+ a c))

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Bertrand Russell, My Philosophical Development, 1959

Russell’s Paradox

Some sets are not members of themselvese.g., set of all Jeffersonians

Some sets are members of themselvese.g., set of all things that are non-Jeffersonian

S = the set of all sets that are not members of themselves

Is S a member of itself?

Russell’s Paradox

• S = set of all sets that are not members of themselves

• Is S a member of itself?

Russell’s Paradox

• S = set of all sets that are not members of themselves

• Is S a member of itself?– If S is an element of S, then S is a member of

itself and should not be in S.– If S is not an element of S, then S is not a

member of itself, and should be in S.

Ban Self-Reference?

• Principia Mathematica attempted to resolve this paragraph by banning self-reference

• Every set has a type– The lowest type of set can contain only “objects”,

not “sets”– The next type of set can contain objects and sets

of objects, but not sets of sets

Russell’s Resolution (?)

Set ::= Setn

Set0 ::= { x | x is an Object }

Setn ::= { x | x is an Object or a Setn - 1 }

S: Setn

Is S a member of itself?No, it is a Setn so, it can’t be a member of a Setn

Epimenides Paradox

Epidenides (a Cretan): “All Cretans are liars.”

Equivalently:“This statement is false.”

Russell’s types can help with the set paradox, but not with these.

Gödel’s “Solution”

All consistent axiomatic formulations of number theory include undecidable propositions.

undecidable: cannot be proven either true or false inside the system.

Kurt Gödel• Born 1906 in Brno (now

Czech Republic, then Austria-Hungary)

• 1931: publishes Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme (On Formally Undecidable Propositions of Principia Mathematica and Related Systems)

The Information, Chapter 6

1939: flees ViennaInstitute for Advanced

Study, PrincetonDied in 1978 –

convinced everything was poisoned and refused to eat

Charge

Today:Incompleteness: there are theorems that cannot be proven

MondayUncomputability: there are problems that cannot be solved by any algorithm

Wednesday: PS7 Due