Steeple #2: Gödel s First Incompleteness Theorem (and thereafter: Steeple #3: Determining Whether a Machine Can Match Gödel)

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1 Steeple #2: Gödel s First Incompleteness Theorem (and thereafter: Steeple #3: Determining Whether a Machine Can Match Gödel) Selmer Bringsjord Are Humans Rational? v RPI Troy NY USA

2 The Power of Study

3 History Made in Milliseconds

4 History Made in Milliseconds

5 History Made in Milliseconds

6 History Made in Milliseconds

7 History Made in Milliseconds

8 History Made in Milliseconds

9 History Made in Milliseconds

10 Thanks to the NFL Films wiring the Patriots assistant coaches for sound in that crazy final sequence for Belichick s team, there is audiovisual evidence of them subbing Butler for linebacker Akeem Ayers as they realized the Seahawks were going three wide in the huddle. Carroll admitted the Seahawks had called that pass play with success in the past, and you can bet the Patriots had seen it and welcomed preparing for it again. It was clear Butler had studied it well, with how quickly he jumped the route while Wilson was trying to connect with Lockette.

Thanks to the NFL Films wiring the Patriots assistant coaches for sound in that crazy final sequence for Belichick s team, there is audiovisual evidence of them subbing Butler for linebacker Akeem

11 Thanks to the NFL Films wiring the Patriots assistant coaches for sound in that crazy final sequence for Belichick s team, there is audiovisual evidence of them subbing Butler for linebacker Akeem Ayers as they realized the Seahawks were going three wide in the huddle. Carroll admitted the Seahawks had called that pass play with success in the past, and you can bet the Patriots had seen it and welcomed preparing for it again. It was clear Butler had studied it well, with how quickly he jumped the route while Wilson was trying to connect with Lockette.

12 Thanks to the NFL Films wiring the Patriots assistant coaches for sound in that crazy final sequence for Belichick s team, there is audiovisual evidence of them subbing Butler for linebacker Akeem Ayers as they realized the Seahawks were going three wide in the huddle. Carroll admitted the Seahawks had called that pass play with success in the past, and you can bet the Patriots had seen it and welcomed preparing for it again. It was clear Butler had studied it well, with how quickly he jumped the route while Wilson was trying to connect with Lockette.

13 Thanks to the NFL Films wiring the Patriots assistant coaches for sound in that crazy final sequence for Belichick s team, there is audiovisual evidence of them subbing Butler for linebacker Akeem Ayers as they realized the Seahawks were going three wide in the huddle. Carroll admitted the Seahawks had called that pass play with success in the past, and you can bet the Patriots had seen it and welcomed preparing for it again. It was clear Butler had studied it well, with how quickly he jumped the route while Wilson was trying to connect with Lockette.

14 Thanks to the NFL Films wiring the Patriots assistant coaches for sound in that crazy final sequence for Belichick s team, there is audiovisual evidence of them subbing Butler for linebacker Akeem Ayers as they realized the Seahawks were going three wide in the huddle. Carroll admitted the Seahawks had called that pass play with success in the past, and you can bet the Patriots had seen it and welcomed preparing for it again. It was clear Butler had studied it well, with how quickly he jumped the route while Wilson was trying to connect with Lockette.

15 Thanks to the NFL Films wiring the Patriots assistant coaches for sound in that crazy final sequence for Belichick s team, there is audiovisual evidence of them subbing Butler for linebacker Akeem Ayers as they realized the Seahawks were going three wide in the huddle. Carroll admitted the Seahawks had called that pass play with success in the past, and you can bet the Patriots had seen it and welcomed preparing for it again. It was clear Butler had studied it well, with how quickly he jumped the route while Wilson was trying to connect with Lockette.

16 The End Result of Study

17 The End Result of Study

18 Steeple #2 (Incompleteness)

19 Goal: Put you in position to prove Gödel s first incompleteness theorem!

20 Goal: Put you in position to prove Gödel s first incompleteness theorem! We have the background (if).

21 The Liar Tree

22 The Liar Tree The Liar Paradox

23 The Liar Tree The Liar Paradox

24 The Liar Tree The Liar Paradox Pure Proof-Theoretic Route

25 The Liar Tree The Liar Paradox Pure Proof-Theoretic Route

26 The Liar Tree The Liar Paradox Pure Proof-Theoretic Route Via Tarski s Theorem

27 The Liar Tree The Liar Paradox Pure Proof-Theoretic Route Via Tarski s Theorem

28 The Liar Tree The Liar Paradox Pure Proof-Theoretic Route Via Tarski s Theorem Ergo, step one: What is LP?

29 Remember The (Economical) Liar which will be our guide!

30 Remember The (Economical) Liar which will be our guide! L: This sentence is false.

31 Remember The (Economical) Liar which will be our guide! L: This sentence is false. Suppose that T(L); then T(L).

32 Remember The (Economical) Liar which will be our guide! L: This sentence is false. Suppose that T(L); then T(L). Suppose that T(L) then T(L).

33 Remember The (Economical) Liar which will be our guide! L: This sentence is false. Suppose that T(L); then T(L). Suppose that T(L) then T(L). Hence: T(L) iff (i.e., if & only if) T(L).

34 Remember The (Economical) Liar which will be our guide! L: This sentence is false. Suppose that T(L); then T(L). Suppose that T(L) then T(L). Hence: T(L) iff (i.e., if & only if) T(L). Contradiction!

35 Next, recall: PA (Peano Arithmetic): A1 x(0 = s(x)) A2 x y(s(x) = s(y) x = y) A3 x(x = 0 y(x = s(y)) A4 x(x +0=x) A5 x y(x + s(y) =s(x + y)) A6 x(x 0 = 0) A7 x y(x s(y) =(x y)+x) And, every sentence that is the universal closure of an instance of ([ (0) x( (x) (s(x))] x (x)) where (x) is open w with variable x, and perhaps others, free.

36 Arithmetic is Part of All Things Sci/Eng/Tech! and courtesy of Gödel: We can t even prove all truths of arithmetic! Each circle is a larger part of the formal sciences. PA

37 Gödel Numbering

38 Gödel Numbering Problem: How do we make a formula refer to other formula and itself (and also other objects like proofs, terms etc.)

39 Gödel Numbering Problem: How do we make a formula refer to other formula and itself (and also other objects like proofs, terms etc.) Solution: Gödel numbering

40 Gödel Numbering Problem: How do we make a formula refer to other formula and itself (and also other objects like proofs, terms etc.) Solution: Gödel numbering! f(x, a)

41 Gödel Numbering Problem: How do we make a formula refer to other formula and itself (and also other objects like proofs, terms etc.) Solution: Gödel numbering! f(x, a) Syntactic objects

42 Gödel Numbering Problem: How do we make a formula refer to other formula and itself (and also other objects like proofs, terms etc.) Solution: Gödel numbering! f(x, a) Syntactic objects (formulae, programs, terms, proofs etc)

43 Gödel Numbering Problem: How do we make a formula refer to other formula and itself (and also other objects like proofs, terms etc.) Solution: Gödel numbering! f(x, a) Syntactic objects (formulae, programs, terms, proofs etc)

44 Gödel Numbering Problem: How do we make a formula refer to other formula and itself (and also other objects like proofs, terms etc.) Solution: Gödel numbering 323! f(x, a) Syntactic objects (formulae, programs, terms, proofs etc)

45 Gödel Numbering Problem: How do we make a formula refer to other formula and itself (and also other objects like proofs, terms etc.) Solution: Gödel numbering 323! f(x, a) Syntactic objects Gödel number (formulae, programs, terms, proofs etc)

46 Gödel Numbering Problem: How do we make a formula refer to other formula and itself (and also other objects like proofs, terms etc.) Solution: Gödel numbering 323! f(x, a) Syntactic objects Gödel number (formulae, programs, terms, proofs etc)

47 Gödel Numbering Problem: How do we make a formula refer to other formula and itself (and also other objects like proofs, terms etc.) Solution: Gödel numbering! f(x, a) Syntactic objects (formulae, programs, terms, proofs etc) Gödel number

48 Gödel Numbering Problem: How do we make a formula refer to other formula and itself (and also other objects like proofs, terms etc.) Solution: Gödel numbering! f(x, a) Syntactic objects (formulae, programs, terms, proofs etc) Gödel number Gödel numeral

49 Gödel Numbering Problem: How do we make a formula refer to other formula and itself (and also other objects like proofs, terms etc.) Solution: Gödel numbering! f(x, a) Syntactic objects (formulae, programs, terms, proofs etc) Gödel number Gödel numeral back to syntax

50 Gödel Numbering Problem: How do we make a formula refer to other formula and itself (and also other objects like proofs, terms etc.) Solution: Gödel numbering! f(x, a) Syntactic objects (formulae, programs, terms, proofs etc) Gödel number Gödel numeral back to syntax

51 Gödel Numbering Problem: How do we make a formula refer to other formula and itself (and also other objects like proofs, terms etc.) Solution: Gödel numbering! f(x, a) Syntactic objects (formulae, programs, terms, proofs etc) Gödel number n Gödel numeral back to syntax

52 Gödel Numbering Problem: How do we make a formula refer to other formula and itself (and also other objects like proofs, terms etc.) Solution: Gödel numbering! f(x, a) Syntactic objects (formulae, programs, terms, proofs etc) Gödel number Gödel numeral back to syntax n ˆn (or just )

53 Gödel Numbering, the Easy Way Rely on the same ordering scheme used for Steeple #1: Just realize that every entry in a dictionary is named by a number n, and by the same basic lexicographic ordering, every computer program, formula, etc. is named by a number m in a lexicographic ordering going from 1, to 2, to

Gödel Numbering, the Easy Way Rely on the same ordering scheme used for Steeple #1: Just realize that every entry in a dictionary is named by a number n, and by the same basic lexicographic ordering,

54 Gödel Numbering, the Easy Way Rely on the same ordering scheme used for Steeple #1: Just realize that every entry in a dictionary is named by a number n, and by the same basic lexicographic ordering, every computer program, formula, etc. is named by a number m in a lexicographic ordering going from 1, to 2, to So, gimcrack is named by some positive integer k. Hence, I can just refer to this word as k.

55 Gödel s First Incompleteness Theorem Suppose that elementary arithmetic (i.e., PA) is consistent (no contradiction can be derived in it) and program-decidable (there s a program P that, given as input an arbitrary formula p, can decide whether or not p is in PA). Then there is sentence g* in the language of elementary arithmetic which is such that: g* can t be proved from PA (i.e., not PA - g*)! And, not-g* can t be proved from PA either (i.e., not PA - not-g*)!

56 Gödel s First Incompleteness Theorem Suppose that elementary arithmetic (i.e., PA) is consistent (no contradiction can be derived in it) and program-decidable (there s a program P that, given as input an arbitrary formula p, can decide whether or not p is in PA). Then there is sentence g* in the language of elementary arithmetic which is such that: g* can t be proved from PA (i.e., not PA - g*)! And, not-g* can t be proved from PA either (i.e., not PA - not-g*)! (Oh, and: g* is true!)

57 Proof Kernel for Theorem GI Let q(x) be an arbitrary formula of arithmetic with one open variable x. (E.g., x + 3 = 5. And here q(2) would be = 5.) Gödel invented a recipe R that, given any q(x) as an ingredient template that you are free to choose, produces a self-referential formula g such that: PA - g <=> q( g ) Part 1: Recipe R (i.e., a formula g that says: I have property q! )

58 Proof Kernel for Theorem GI Part 2: Follow Recipe R, Guided by The Liar First, for q(x) we choose a formula q* that holds of any s if and only if s can be proved from PA; i.e., PA - q*( s ) iff PA - s (1) Next, we follow Gödel s Recipe R to build a g* such that: PA - g* <=> not-q*( g* ) (2)

59 Proof Kernel for Theorem GI Part 2: Follow Recipe R, Guided by The Liar First, for q(x) we choose a formula q* that holds of any s if and only if s can be proved from PA; i.e., PA - q*( s ) iff PA - s (1) Next, we follow Gödel s Recipe R to build a g* such that: PA - g* <=> not-q*( g* ) (2) g* thus says: I m not true! / I m not provable. And so, the key question (assignment!): PA - g*?!?

60 Austin Hardyesque Indirect Proof GI: g* isn t provable from PA; nor is the negation of g*! Proof: Let s follow The Liar: Suppose that g* is provable from PA; i.e., suppose PA - g*. Then by (1), with g* substituted for s, we have: PA - q*( g* ) iff PA - g* (1 ) From our supposition and working right to left by modus ponens on (1 ) we deduce: PA - q*( g* ) (3.1) But from our supposition and the earlier (see previous slide) (2), we can deduce by modus ponens that from PA the opposite can be proved! I.e., we have: PA - not-q*( g* ) (3.2) But (3.1) and (3.2) together means that PA is inconsistent, since it generates a contradiction. Hence by indirect proof g* is not provable from PA.

61 Austin Hardyesque Indirect Proof GI: g* isn t provable from PA; nor is the negation of g*! Proof: Let s follow The Liar: Suppose that g* is provable from PA; i.e., suppose PA - g*. Then by (1), with g* substituted for s, we have: PA - q*( g* ) iff PA - g* (1 ) From our supposition and working right to left by modus ponens on (1 ) we deduce: PA - q*( g* ) (3.1) But from our supposition and the earlier (see previous slide) (2), we can deduce by modus ponens that from PA the opposite can be proved! I.e., we have: PA - not-q*( g* ) (3.2) But (3.1) and (3.2) together means that PA is inconsistent, since it generates a contradiction. Hence by indirect proof g* is not provable from PA. What about the second option? Can you follow The Liar to show that supposing that the negation of g* (i.e., not-g*) is provable from PA also leads to a contradiction, and hence can t be? (Test 3 A+ question.)

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64 Finally: Could an AI Ever Match Gödel?

65 Actually, thanks to AFOSR: GI (& GT): Done...

66 Licato, J.; Govindarajulu, N.; Bringsjord, S.; Pomeranz, M.; Gittelson, L Analogico- Deductive Generation of Godel s First Incompleteness Theorem from the Liar Paradox. In Proceedings of IJCAI Pdf Govindarajulu, N.; Licato, J.; Bringsjord, S Small Steps Toward Hypercomputation via Infinitary Machine Proof Verification and Proof Generation. In Proceedings of UCNC Pdf

67 Licato, J.; Govindarajulu, N.; Bringsjord, S.; Pomeranz, M.; Gittelson, L Analogico- Deductive Generation of Godel s First Incompleteness Theorem from the Liar Paradox. In Proceedings of IJCAI Pdf Govindarajulu, N.; Licato, J.; Bringsjord, S Small Steps Toward Hypercomputation via Infinitary Machine Proof Verification and Proof Generation. In Proceedings of UCNC Pdf

68 Continuum of Results Liar Paradox (L) Collection of semiformal statements Syntax not rigorously defined Represents intuitive understanding of problem domain s = This statement is a lie There is a statement that is neither true nor false.... G1 Completely formal statements Syntax very rigorously defined Purely mathematical objects: numbers, formal theories, etc. s =? φ LA (PA φ) (PA φ)...

69 Continuum of Results Liar Paradox (L) Collection of semiformal statements Syntax not rigorously defined Represents intuitive understanding of problem domain s = This statement is a lie There is a statement that is neither true nor false.... G1 Completely formal statements Syntax very rigorously defined Purely mathematical objects: numbers, formal theories, etc. s =? φ LA (PA φ) (PA φ)...

70 Continuum of Results Liar Paradox (L) Collection of semiformal statements Syntax not rigorously defined Represents intuitive understanding of problem domain s = This statement is a lie There is a statement that is neither true nor false.... S Semiformal statements (but more formal than L) Syntax somewhat rigorously defined Somewhat intuitive; deals with stories of reasoners and utterances made by inhabitants of an island s = Bs p Bp B p... G1 Completely formal statements Syntax very rigorously defined Purely mathematical objects: numbers, formal theories, etc. s =? φ LA (PA φ) (PA φ)...

71 Continuum of Results Liar Paradox (L) Collection of semiformal statements Syntax not rigorously defined Represents intuitive understanding of problem domain s = This statement is a lie There is a statement that is neither true nor false.... S Semiformal statements (but more formal than L) Syntax somewhat rigorously defined Somewhat intuitive; deals with stories of reasoners and utterances made by inhabitants of an island s = Bs p Bp B p... G1 Completely formal statements Syntax very rigorously defined Purely mathematical objects: numbers, formal theories, etc. s =? φ LA (PA φ) (PA φ)...

72 Continuum of Results Liar Paradox (L) Collection of semiformal statements Syntax not rigorously defined Represents intuitive understanding of problem domain s = This statement is a lie There is a statement that is neither true nor false.... S Semiformal statements (but more formal than L) Syntax somewhat rigorously defined Somewhat intuitive; deals with stories of reasoners and utterances made by inhabitants of an island s = Bs p Bp B p... G1 Completely formal statements Syntax very rigorously defined Purely mathematical objects: numbers, formal theories, etc. s =? φ LA (PA φ) (PA φ)...

73 Continuum of Results Liar Paradox (L) Collection of semiformal statements Syntax not rigorously defined Represents intuitive understanding of problem domain s = This statement is a lie There is a statement that is neither true nor false.... S Semiformal statements (but more formal than L) Syntax somewhat rigorously defined Somewhat intuitive; deals with stories of reasoners and utterances made by inhabitants of an island Done s = Bs p Bp B p... G1 Completely formal statements Syntax very rigorously defined Purely mathematical objects: numbers, formal theories, etc. s =? φ LA (PA φ) (PA φ)...

74 Scare Quotes Because, Yet, In The End, Still Humbling

75 Scare Quotes Because, Yet, In The End, Still Humbling 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving....

76 Scare Quotes Because, Yet, In The End, Still Humbling 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals.

77 Scare Quotes Because, Yet, In The End, Still Humbling 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false.

78 Scare Quotes Because, Yet, In The End, Still Humbling 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false.

79 Scare Quotes Because, Yet, In The End, Still Humbling 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false.

80 Scare Quotes Because, Yet, In The End, Still Humbling 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false.

81 Scare Quotes Because, Yet, In The End, Still Humbling 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is is true.)

82 Scare Quotes Because, Yet, In The End, Still Humbling 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is is true.)

83 Scare Quotes Because, Yet, In The End, Still Humbling 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is is true.) Silver Blaze

84 Scare Quotes Because, Yet, In The End, Still Humbling 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is is true.) Silver Blaze

85 Scare Quotes Because, Yet, In The End, Still Humbling 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is is true.) Silver Blaze Straker dead from a savage blow. Inspector Gregory: Fitzroy Simpson definitely guilty, and we arrested him!

86 Scare Quotes Because, Yet, In The End, Still Humbling 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is is true.) Silver Blaze Straker dead from a savage blow. Inspector Gregory: Fitzroy Simpson definitely guilty, and we arrested him! Holmes: But I ve created a scenario in which all your clues/ evidence are true, but Fitzroy is perfectly innocent

87 Scare Quotes Because, Yet, In The End, Still Humbling 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is is true.) Silver Blaze Straker dead from a savage blow. Inspector Gregory: Fitzroy Simpson definitely guilty, and we arrested him! Holmes: But I ve created a scenario in which all your clues/ evidence are true, but Fitzroy is perfectly innocent Gödel: I ve created a scenario where all of ZFC is true, but so is the Continuum Hypothesis!

88 Scare Quotes Because, Yet, In The End, Still Humbling 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is is true.) Silver Blaze Straker dead from a savage blow. Inspector Gregory: Fitzroy Simpson definitely guilty, and we arrested him! Holmes: But I ve created a scenario in which all your clues/ evidence are true, but Fitzroy is perfectly innocent Gödel: I ve created a scenario where all of ZFC is true, but so is the Continuum Hypothesis!

89 Scare Quotes Because, Yet, In The End, Still Humbling 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is is true.) Silver Blaze Straker dead from a savage blow. Inspector Gregory: Fitzroy Simpson definitely guilty, and we arrested him! Holmes: But I ve created a scenario in which all your clues/ evidence are true, but Fitzroy is perfectly innocent Gödel: I ve created a scenario where all of ZFC is true, but so is the Continuum Hypothesis!

90 Main Claim

91 Main Claim

92 And Supporting Main Claim

94 slutten

Gödel s First Incompleteness Theorem (excerpted from Gödel s Great Theorems) Selmer Bringsjord Intro to Logic May RPI Troy NY USA

Gödel s First Incompleteness Theorem (excerpted from Gödel s Great Theorems) Selmer Bringsjord Intro to Logic May 2 2016 RPI Troy NY USA Thursday: Can a machine match Gödel? Grade roundup (not today; let