Remarks on the Automated Discovery of Gödel’s Incompleteness Theorems
Selmer BringsjordRensselaer AI & Reasoning (RAIR) Laboratory
Department of Cognitive ScienceDepartment of Computer Science
Rensselaer Polytechnic Institute (RPI)Troy NY 12180 USA
[email protected] Arlington VA
Remarks on the Automated Discovery of Gödel’s Incompleteness Theorems
Selmer BringsjordRensselaer AI & Reasoning (RAIR) Laboratory
Department of Cognitive ScienceDepartment of Computer Science
Rensselaer Polytechnic Institute (RPI)Troy NY 12180 USA
[email protected] Arlington VA
The support of this work by IARPA (A-SpaceX) and NSF (CreativeIT) is acknowledged with deep gratitude.
Gödel, God, and Golems:Gödelian Essays on Minds and Machines
• Failed Arguments from Gödelian Incompleteness to the Falsity of Mechanism
• Minds & Machines
• Victorious Argument from Gödelian Incompleteness to the Falsity of Mechanism
• Theoretical Computer Science
• On Gödelian Arguments Against Computationalism from Ascension Through Time of Human Intelligence
• Applied Mathematics and Computation
• On Machines Discovering Gödelian Incompleteness Results
• Gödel’s Modal Argument for God’s Existence
Gödel, God, and Golems:Gödelian Essays on Minds and Machines
• Failed Arguments from Gödelian Incompleteness to the Falsity of Mechanism
• Minds & Machines
• Victorious Argument from Gödelian Incompleteness to the Falsity of Mechanism
• Theoretical Computer Science
• On Gödelian Arguments Against Computationalism from Ascension Through Time of Human Intelligence
• Applied Mathematics and Computation
• On Machines Discovering Gödelian Incompleteness Results
• Gödel’s Modal Argument for God’s Existence
I’m very skeptical.
Two reasons:
Background: AR{f |f : N → N}
(Information Processing)
Background: AR{f |f : N → N}
Turing Limit
(Information Processing)
Background: AR{f |f : N → N}
Turing Limit
H(n, k, u, v)
(Information Processing)
Background: AR{f |f : N → N}
Turing Limit
H(n, k, u, v)
(Information Processing)
(chess, swimming, flying, locomotion)
Background: AR{f |f : N → N}
Turing Limit
H(n, k, u, v)∃kH(n, k, u, v)
(Information Processing)
(chess, swimming, flying, locomotion)
Background: AR{f |f : N → N}
Turing Limit
H(n, k, u, v)∃kH(n, k, u, v)
Φ ! φ?Σ1
(Information Processing)
(chess, swimming, flying, locomotion)
Background: AR{f |f : N → N}
Turing Limit
H(n, k, u, v)∃kH(n, k, u, v)
∀u∀v[∃kH(n, k, u, v) ↔ ∃k′H(m, k′, u, v)]Π2
Φ ! φ?Σ1
(Information Processing)
(chess, swimming, flying, locomotion)
Superminds (2003)
Turing Limit
Information Processing
Phenomena that can’t be expressed in any third-person scheme
persons
animals
People Harness Hypercomputation, and More
29
by
SUPERMINDSPeople Harness Hypercomputation, and More
bySelmer Bringsjord and Micael Zenzen
This is the first book-length presentation and defense of a new theory of human andmachine cognition, according to which human persons are superminds. Superminds arecapable of processing information not only at and below the level of Turing machines(standard computers), but above that level (the “Turing Limit”), as information processingdevices that have not yet been (and perhaps can never be) built, but have beenmathematically specified; these devices are known as super-Turing machines orhypercomputers. Superminds, as explained herein, also have properties no machine,whether above or below the Turing Limit, can have. The present book is the third andpivotal volume in Bringsjord’s supermind quartet; the first two books were What RobotsCan and Can’t Be (Kluwer) and AI and Literary Creativity (Lawrence Erlbaum). The finalchapter of this book offers eight prescriptions for the concrete practice of AI and cognitivescience in light of the fact that we are superminds.
SELMER
BR
ING
SJOR
D
AN
D M
ICH
AEL ZEN
ZENSU
PERM
IND
SPeople H
arness Hypercom
putation, and More
KLUWER ACADEMIC PUBLISHERS COGS 29
Bringsjord COGS 29 PB(2)xpr 07-02-2003 16:26 Pagina 1
Superminds (2003)
Turing Limit
Information Processing
Phenomena that can’t be expressed in any third-person scheme
persons
animals (chess, swimming, flying, locomotion)
People Harness Hypercomputation, and More
29
by
SUPERMINDSPeople Harness Hypercomputation, and More
bySelmer Bringsjord and Micael Zenzen
This is the first book-length presentation and defense of a new theory of human andmachine cognition, according to which human persons are superminds. Superminds arecapable of processing information not only at and below the level of Turing machines(standard computers), but above that level (the “Turing Limit”), as information processingdevices that have not yet been (and perhaps can never be) built, but have beenmathematically specified; these devices are known as super-Turing machines orhypercomputers. Superminds, as explained herein, also have properties no machine,whether above or below the Turing Limit, can have. The present book is the third andpivotal volume in Bringsjord’s supermind quartet; the first two books were What RobotsCan and Can’t Be (Kluwer) and AI and Literary Creativity (Lawrence Erlbaum). The finalchapter of this book offers eight prescriptions for the concrete practice of AI and cognitivescience in light of the fact that we are superminds.
SELMER
BR
ING
SJOR
D
AN
D M
ICH
AEL ZEN
ZENSU
PERM
IND
SPeople H
arness Hypercom
putation, and More
KLUWER ACADEMIC PUBLISHERS COGS 29
Bringsjord COGS 29 PB(2)xpr 07-02-2003 16:26 Pagina 1
Th 89, and how Selmer knows that those claiming any such thing as that they have a system able to semi-/automatically discover Gödel’s incompleteness theorems (GI and GII) are, well, ...
Consider Theorem 89 (Th 89) from Patrick Suppes’ textbook Axiomatic Set Theory, penned in 1960 to introduce set theory to its readers. Th 89 says simply that the power set of the null set is the set composed of just the null set:
P(∅) = {∅}
Suppes gives a rather short proof of this theorem, to wit:
Since ∅ ⊆ ∅, ∅ ∈ P(∅).
but then by Theorem 4 A = ∅. QEDMoreoever, since if A ∈ P(∅), then by Theorem 86 A ⊆ ∅,
The greatest automated theorem prover (ATP) on the planet as of 2006, Vampire, cannot discover a proof of Th 89. (Now? Meta-Prover?)
This was brought to my attention by Konstantine (Kostas) Arkoudas. When he first told me, I was, frankly, very skeptical: After all, Th 89 is so simple that it’s reminiscent of 1956-level “triumphs.” I decided to first prove Th 89 from scratch, qua human, not machine. My proof, which appears on the following two slides, is much more explicit than Suppes’, and has a lot more “human-natural” structure.
So I had proved the theorem from scratch, using only ingredients that would be given to Vampire.
How does Vampire fare? Well, let’s look at a little Athena script written by Konstantine (Kostas) Arkoudas, in which we’ll compare it to the efficacy of the the human-natural approach...
# The following fails:
(!derive goal premises)
# So we need to do some more work:
(define lemma (forall ?x ?y (if (subset ?x (singleton ?y)) (or (= ?x null) (= ?x (singleton ?y))))))
# The lemma is easy:
(!derive lemma premises)
# We now break the goal into 2 halves:
(define goal-1 (subset (pset (singleton null)) (pair null (singleton null))))
(define goal-2 (subset (pair null (singleton null)) (pset (singleton null))))
# Both are easily derivable:
(!prove goal-1)
(!prove goal-2)
# And now the original goal can be proved:
(!prove goal)
Theorem: (forall ?x:Set (forall ?y:Set (if (subset ?x (singleton ?y)) (or (= ?x null) (= ?x (singleton ?y))))))
Review of GI & GII ...
Sar := {+, ·, 0, 1}
Assume first-order logic.
Assume understanding of consistency with respect to a set of first-order formulas...
Assume understanding of Turing machine-level decidability...
Assume understanding of Turing machine-level computability...
an Sar − formula φ(v0, . . . , vr−1) such that for allRelation R ⊂ N r is representable in Φ if there is
no, . . . , nr−1 ∈ N :
If Rn0 . . . nr−1 then Φ ! φ(n̄o, . . . , n̄r−1);
If not Rn0 . . . nr−1 then Φ ! ¬φ(n̄o, . . . , n̄r−1).
Representability (relations)
Function F : N r −→ N is representable in Φ if there is
an Sar − formula φ(v0, . . . , vr−1, vr) such that for allno, . . . , nr−1, nr ∈ N :
If F (n0 . . . nr−1) = nr then Φ ! φ(n̄o, . . . , n̄r−1, n̄r);If F (n0 . . . nr−1) != nr then Φ " ¬φ(n̄o, . . . , n̄r−1, n̄r);
Φ ! ∃=1vrφ(n̄o, . . . , n̄r−1, n̄r).
Representability (functions)
We say that Φ allows representations if
all decidable relations and all computable functions over N are representable in Φ.
Gödel’s First Incompleteness Theorem
Let Φ be consistent and decidable and suppose also that
Φ allows representations. Then there is an Sar-sentence φ
such that neither Φ ! φ nor Φ ! ¬φ.
Assume understanding of provability in some proof theory for standard first-order logic...
Assume understanding of Gödel numbering...
Assume a Turing-machine enumeration of all proofs..
Toward Gödel’s Second Incompleteness Theorem...
Hnm iff the mth proof yields φ where n = nφ.
Obviously, Φ ! φ iff there is an m ∈ N s.t. Hnφm.
H can be represented in Φ by a formula φH(v0, v1) ∈ LSar2 .
With x, y for v0, v1, we set DerΦ(x) := ∃yφH(x, y).
Set ConsisΦ := ¬DerΦ( ¯n¬0=0).
Gödel’s Second Incompleteness Theorem
Let Φ be consistent and decidable with Φ ⊂ ΦPA.
Let Φ be consistent and decidable with Φ ⊂ ΦZFC .
Then not Φ ! ConsisΦ.
In other words, as many people put it: “You cannot prove that mathematics is free of contradiction using (classical) mathematics.” I would rather more circumspectly say: “Using ordinary mechanical methods of proof specification, a machine can’t specify a proof that mathematics is consistent, where ‘mathematics is consistent’ is encoded as above. The same holds for a human.”
Players; Narrowing the Field
• Ammon (1993)
• Quaife (1988)
• Sieg & Field (2005)
• Shankar (1994)
“Eliminating” Ammon
Oh boy. His self-appraisal makes the sanguinity of Newell and Simon seem like clinical depression.
perfect match with Kleene’s proof (Intro to Metamathematics)
“implicitly rediscovered Cantor’s diagonal method”
Players; Narrowing the Field
• Ammon (1993)
• Quaife (1988)
• Sieg & Field (2005)
• Shankar (1994)
Players; Narrowing the Field
• Ammon (1993)
• Quaife (1988)
• Sieg & Field (2005)
• Shankar (1994)
“Eliminating” Quaife“[I]t is very difficult to understand how the syntactic context of axioms, theorems and assumptions directs the search in a way that is motivated by the leading ideas of the mathematical subject. The proofs use in every case ‘axioms and previously proven theorems’ in addition to the standard hypotheses for the theorem under consideration. It is clear that the ‘previously proven theorems’ are strategically selected, and it is fair to ask, whether the full proof — from axioms through intermediate results to the meta-mathematical theorems — should be viewed as ‘automated’ or rather as ‘interactive’ with automated large logical steps.”
—Sieg & Field (2005)
“Eliminating” Quaife
“Eliminating” Quaife
“Eliminating” Quaife
Players; Narrowing the Field
• Ammon (1993)
• Quaife (1988)
• Sieg & Field (2005)
• Shankar (1994)
Players; Narrowing the Field
• Ammon (1993)
• Quaife (1988)
• Sieg & Field (2005)
• Shankar (1994)
Comments on Sieg & Field ...
Confessed Quaife-ishness
“Definitional and other mathematical equivalences are used to obtain either a new available formula from which the current goal is extractable or to get an equivalent statement as a new goal. This we would like to do relative to a developing background theory; currently, we just add the definitions and lemmata explicitly to the list of premises.”
—Sieg & Field p. 325
Proof of Non-Provability of G
Proof of Non-Provability of G
For the second part, omega-consistency assumed, but similarly simple.
Suggestion:
Define discovery...
