Could an AI Ever Match Gödel? (excerpted from Gödel s Great Theorems) Selmer Bringsjord Intro to Logic May RPI Troy NY USA
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1 Could an AI Ever Match Gödel? (excerpted from Gödel s Great Theorems) Selmer Bringsjord Intro to Logic May RPI Troy NY USA
2 Played last night :)?
3 Played last night :)? If you won, please remember the Selmerand-Kevin fund for presenting our paper on why you shouldn t have played :) (If you lost, you shouldn t have played, as we show in said paper.)
4 Contest Winners
5 Contest Winners 3rd: Louis Hyde
6 Contest Winners 2nd: Mark Robinson 3rd: Louis Hyde
7 Contest Winners 1st: Eamon Olive 2nd: Mark Robinson 3rd: Louis Hyde
8 Contest Winners 1st: Eamon Olive 2nd: Mark Robinson 3rd: Louis Hyde
9
10
11 L Hyde Proof
12 L Hyde Proof
13 Logistics
14 Logistics 1. Student researchers for summer: we ve gotta get something in today, apparently. See me (preferably in office hrs).
15 Logistics 1. Student researchers for summer: we ve gotta get something in today, apparently. See me (preferably in office hrs). 2. Last of adjusted T2s here (for pickup after).
16 Logistics 1. Student researchers for summer: we ve gotta get something in today, apparently. See me (preferably in office hrs). 2. Last of adjusted T2s here (for pickup after). 3. Need CD#s from some who are signed up & receiving s. I ed you. Come up after to provide on sheets.
17 Logistics 1. Student researchers for summer: we ve gotta get something in today, apparently. See me (preferably in office hrs). 2. Last of adjusted T2s here (for pickup after). 3. Need CD#s from some who are signed up & receiving s. I ed you. Come up after to provide on sheets. 4. Must be on official list to take Test 3 & receive a grade in class. If you haven t received the recent on the contest winners, you re not signed up!
18 Logistics 1. Student researchers for summer: we ve gotta get something in today, apparently. See me (preferably in office hrs). 2. Last of adjusted T2s here (for pickup after). 3. Need CD#s from some who are signed up & receiving s. I ed you. Come up after to provide on sheets. 4. Must be on official list to take Test 3 & receive a grade in class. If you haven t received the recent on the contest winners, you re not signed up! 5. Test 3 Prep Doc: Watch the web page!
19 Logistics 1. Student researchers for summer: we ve gotta get something in today, apparently. See me (preferably in office hrs). 2. Last of adjusted T2s here (for pickup after). 3. Need CD#s from some who are signed up & receiving s. I ed you. Come up after to provide on sheets. 4. Must be on official list to take Test 3 & receive a grade in class. If you haven t received the recent on the contest winners, you re not signed up! 5. Test 3 Prep Doc: Watch the web page! 5.1.But you already have assignment/challenge from last class.
20 Logistics 1. Student researchers for summer: we ve gotta get something in today, apparently. See me (preferably in office hrs). 2. Last of adjusted T2s here (for pickup after). 3. Need CD#s from some who are signed up & receiving s. I ed you. Come up after to provide on sheets. 4. Must be on official list to take Test 3 & receive a grade in class. If you haven t received the recent on the contest winners, you re not signed up! 5. Test 3 Prep Doc: Watch the web page! 5.1.But you already have assignment/challenge from last class. 5.2.And, consider such S5 formulae as:
21 Logistics 1. Student researchers for summer: we ve gotta get something in today, apparently. See me (preferably in office hrs). 2. Last of adjusted T2s here (for pickup after). 3. Need CD#s from some who are signed up & receiving s. I ed you. Come up after to provide on sheets. 4. Must be on official list to take Test 3 & receive a grade in class. If you haven t received the recent on the contest winners, you re not signed up! 5. Test 3 Prep Doc: Watch the web page! 5.1.But you already have assignment/challenge from last class. 5.2.And, consider such S5 formulae as:
22 Logistics 1. Student researchers for summer: we ve gotta get something in today, apparently. See me (preferably in office hrs). 2. Last of adjusted T2s here (for pickup after). 3. Need CD#s from some who are signed up & receiving s. I ed you. Come up after to provide on sheets. 4. Must be on official list to take Test 3 & receive a grade in class. If you haven t received the recent on the contest winners, you re not signed up! 5. Test 3 Prep Doc: Watch the web page! 5.1.But you already have assignment/challenge from last class. 5.2.And, consider such S5 formulae as: 6. Test 3 may be provided directly by !
23 Logistics 1. Student researchers for summer: we ve gotta get something in today, apparently. See me (preferably in office hrs). 2. Last of adjusted T2s here (for pickup after). 3. Need CD#s from some who are signed up & receiving s. I ed you. Come up after to provide on sheets. 4. Must be on official list to take Test 3 & receive a grade in class. If you haven t received the recent on the contest winners, you re not signed up! 5. Test 3 Prep Doc: Watch the web page! 5.1.But you already have assignment/challenge from last class. 5.2.And, consider such S5 formulae as: 6. Test 3 may be provided directly by ! 7. If you wish to check grades: my office ; after you turn in Test 3 Monday; my office ; Rini s office Tues; or you can me with subject: grades so far.
24 Okay: Can a machine match Gödel?
25 Okay: Can a machine match Gödel? Or:
26 Okay: Can a machine match Gödel? Or: AlphaGo!
27 Okay: Can a machine match Gödel? Or: AlphaGo! So?
28 Okay: Can a machine match Gödel? Or: AlphaGo! So?
29 Actually, thanks to AFOSR: GI (& GT): Done...
30 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
31 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
32 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)...
33 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)...
34 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)...
35 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)...
36 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 p s = Bs... G1 Completely formal statements Syntax very rigorously defined Purely mathematical objects: numbers, formal theories, etc. φ A s =?...
37 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 p s = Bs... G1 Completely formal statements Syntax very rigorously defined Purely mathematical objects: numbers, formal theories, etc. φ A s =?...
38 Regarding a reasoner on Smullyan s Island: S B(k $ e Bk) goal statement G: From Therefore: this and G 9 p the ( Bp ^ B p) G can be informally
39 Regarding a reasoner on Smullyan s Island: S B(k $ e Bk) goal statement G: From Therefore: this and G 9 p the ( Bp ^ B p) G can be informally These statements are analogous to:
40 Regarding a reasoner on Smullyan s Island: S B(k $ e Bk) goal statement G: From Therefore: this and G 9 p the ( Bp ^ B p) G can be informally These statements are analogous to: D 0 8 q (Proves(,q) $9 n Prf (n, numeral(q))) S 0 Con 8 q Proves(,q)! Proves(, q) S 0!Con 9 q (Proves(, apply(, numeral(q))) ^8 n Proves(, not(prf (n, numeral(q))))) G 0 9 ( Proves(, ) ^ Proves(, not( )))
41 C1b'.!x,y ((Proves(Φ,x) % Proves(Φ,if(x,y))) $ Proves(Φ,y)) {C1b'} Assume! Representability of provability.!q (Proves(Φ,apply(ρ,numeral(q))) " Proves(Φ,q)) {Representability of provability} Assume! D'.!q (Proves(Φ,q) " #n Prf(n,numeral(q))) {D'} Assume! Scon'.!q (Proves(Φ,q) $ Proves(Φ,not(q))) {Scon'} Assume! Fixed Point Lemma.!q #Fq Proves(Φ,iff(Fq,apply(q,numeral(Fq)))) {Fixed Point Lemma} Assume! C2a'.!n,q (Prf(n,numeral(q)) $ Proves(Φ,prf(n,numeral(q)))) {C2a'} Assume! not extraction.!q,r (Proves(Φ,iff(q,apply(not(r),numeral(q)))) " Proves(Φ,iff(q,not(apply(r,numeral(q)))))) {not extraction} Assume! C2b'.!n,q ( Prf(n,numeral(q)) $ Proves(Φ,not(prf(n,numeral(q))))) {C2b'} Assume! C3'.!n,q Proves(Φ,if(prf(n,numeral(q)),apply(ρ,numeral(q)))) {C3'} Assume! Sωcon'. #q (Proves(Φ,apply(ρ,numeral(q))) %!n Proves(Φ,not(prf(n,numeral(q))))) {Sωcon'} Assume! iff expansion.!q,r,s (Proves(q,iff(r,not(s))) " (Proves(q,if(r,not(s))) % Proves(q,if(not(s),r)) % Proves(q,if(not(r),s)) % Proves(q,if(s,not(r))))) {iff expansion} Assume! FOL! S'. #k (Proves(Φ,if(k,not(apply(ρ,numeral(k))))) % Proves(Φ,if(not(k),apply(ρ,numeral(k)))) % Proves(Φ,if(apply(ρ,numeral(k)),not(k))) % Proves(Φ,if(not(apply(ρ,numeral(k))),k))) {Fixed Point Lemma,Representability of provability,iff expansion,not extraction} FOL! G. #φ ( Proves(Φ,φ) % Proves(Φ,not(φ))) {C1b',C2a',C2b',C3',D',Fixed Point Lemma,Representability of provability,scon',sωcon',iff expansion,not extraction} Figure 2: A full proof of G1 in Slate, generated through ADR. Note that S 0 is an expanded version of what actually follows from the analogical transfer of S; this was done for better presentability of this image.
42 Overview (compressed!)
43 Overview (compressed!) 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving....
44 Overview (compressed!) 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... CH: 8x[(x R ^ Fin(x))! (Count(x) _ x R)]
45 Overview (compressed!) 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... CH: 8x[(x R ^ Fin(x))! (Count(x) _ x R)] math consistent 1938 Gödel: If Con ZFC, then ZFC 6` CH
46 Overview (compressed!) 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... CH: 8x[(x R ^ Fin(x))! (Count(x) _ x R)] math consistent 1938 Gödel: If Con ZFC, then ZFC 6` CH
47 Overview (compressed!) 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... CH: 8x[(x R ^ Fin(x))! (Count(x) _ x R)] math consistent 1938 Gödel: If Con ZFC, then ZFC 6` CH
48 Overview (compressed!) 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... CH: 8x[(x R ^ Fin(x))! (Count(x) _ x R)] math consistent 1938 Gödel: If Con ZFC, then ZFC 6` CH
49 Overview (compressed!) 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... CH: 8x[(x R ^ Fin(x))! (Count(x) _ x R)] math consistent 1938 Gödel: If Con ZFC, then ZFC 6` CH math consistent 1963 Cohen: If Con ZFC, then ZFC 6` CH
50 Overview (compressed!) 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... CH: 8x[(x R ^ Fin(x))! (Count(x) _ x R)] math consistent 1938 Gödel: If Con ZFC, then ZFC 6` CH math consistent 1963 Cohen: If Con ZFC, then ZFC 6` CH
51 Overview (compressed!) 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... CH: 8x[(x R ^ Fin(x))! (Count(x) _ x R)] math consistent 1938 Gödel: If Con ZFC, then ZFC 6` CH math consistent 1963 Cohen: If Con ZFC, then ZFC 6` CH Silver Blaze
52 Overview (compressed!) 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... CH: 8x[(x R ^ Fin(x))! (Count(x) _ x R)] math consistent 1938 Gödel: If Con ZFC, then ZFC 6` CH math consistent 1963 Cohen: If Con ZFC, then ZFC 6` CH Silver Blaze
53 Overview (compressed!) 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... CH: 8x[(x R ^ Fin(x))! (Count(x) _ x R)] math consistent 1938 Gödel: If Con ZFC, then ZFC 6` CH math consistent 1963 Cohen: If Con ZFC, then ZFC 6` CH Silver Blaze Straker dead from a savage blow. Inspector Gregory: Fitzroy Simpson definitely guilty, and arrested!
54 Overview (compressed!) 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... CH: 8x[(x R ^ Fin(x))! (Count(x) _ x R)] math consistent 1938 Gödel: If Con ZFC, then ZFC 6` CH math consistent 1963 Cohen: If Con ZFC, then ZFC 6` CH Silver Blaze Straker dead from a savage blow. Inspector Gregory: Fitzroy Simpson definitely guilty, and arrested! Holmes: But here s a scenario where all your clues/ evidence are true, but Fitzroy is perfectly innocent
55 Overview (compressed!) 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... CH: 8x[(x R ^ Fin(x))! (Count(x) _ x R)] math consistent 1938 Gödel: If Con ZFC, then ZFC 6` CH math consistent 1963 Cohen: If Con ZFC, then ZFC 6` CH Silver Blaze Straker dead from a savage blow. Inspector Gregory: Fitzroy Simpson definitely guilty, and arrested! Holmes: But here s a scenario where all your clues/ evidence are true, but Fitzroy is perfectly innocent Gödel: But here s a scenario where all of ZFC is true, but so is the Continuum Hypothesis!
56 Overview (compressed!) 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... CH: 8x[(x R ^ Fin(x))! (Count(x) _ x R)] math consistent 1938 Gödel: If Con ZFC, then ZFC 6` CH math consistent 1963 Cohen: If Con ZFC, then ZFC 6` CH Silver Blaze Straker dead from a savage blow. Inspector Gregory: Fitzroy Simpson definitely guilty, and arrested! Holmes: But here s a scenario where all your clues/ evidence are true, but Fitzroy is perfectly innocent Gödel: But here s a scenario where all of ZFC is true, but so is the Continuum Hypothesis!
57 Overview (compressed!) 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... CH: 8x[(x R ^ Fin(x))! (Count(x) _ x R)] math consistent 1938 Gödel: If Con ZFC, then ZFC 6` CH math consistent 1963 Cohen: If Con ZFC, then ZFC 6` CH Silver Blaze Straker dead from a savage blow. Inspector Gregory: Fitzroy Simpson definitely guilty, and arrested! Holmes: But here s a scenario where all your clues/ evidence are true, but Fitzroy is perfectly innocent Gödel: But here s a scenario where all of ZFC is true, but so is the Continuum Hypothesis!
58 Logistics 1. Student researchers for summer: we ve gotta get something in today, apparently. See me (preferably in office hrs). 2. Last of adjusted T2s here (for pickup after). 3. Need CD#s from some who are signed up & receiving s. I ed you. Come up after to provide on sheets. 4. Must be on official list to take Test 3 & receive a grade in class. If you haven t received the recent on the contest winners, you re not signed up! 5. Test 3 Prep Doc: Watch the web page! 5.1.But you already have assignment/challenge from last class. 5.2.And, consider such formulae as: 6. Test 3 may be provided directly by ! 7. If you wish to check grades: my office ; after you turn in Test 3 Monday; my office ; Rini s office Tues; or you can me with subject: grades so far.
59
60 finis
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