gödel’s completeness theoremkryten.mm.rpi.edu/courses/ahr/godel_completeness_part2.pdf ·...

Rensselaer AI & Reasoning (RAIR) LabDepartment of Cognitive ScienceDepartment of Computer Science

Lally School of Management & TechnologyRensselaer Polytechnic Institute (RPI)

Troy, New York 12180 USA

Intro to Logic11/25/2019

Selmer Bringsjord

Gödel’s Completeness Theorem

Rensselaer AI & Reasoning (RAIR) LabDepartment of Cognitive ScienceDepartment of Computer Science

Lally School of Management & TechnologyRensselaer Polytechnic Institute (RPI)

Troy, New York 12180 USA

Intro to Logic11/25/2019

Selmer Bringsjord

Gödel’s Completeness Theorem

Only steeples of rationalism!

Background Context …

Gödel’s Great Theorems (OUP)by Selmer Bringsjord

• Introduction (“The Wager”)

• Brief Preliminaries (e.g. the propositional calculus & FOL)

• The Completeness Theorem

• The First Incompleteness Theorem

• The Second Incompleteness Theorem

• The Speedup Theorem

• The Continuum-Hypothesis Theorem

• The Time-Travel Theorem

• Gödel’s “God Theorem”

• Could a Machine Match Gödel’s Genius?

Some Timeline Points


1906 Brünn, Austria-Hungary


1906 Brünn, Austria-Hungary1923 Vienna



Undergrad in seminar by Schlick



Undergrad in seminar by Schlick1929 Doctoral Dissertation: Proof of Completeness Theorem




1930 Announces (First) Incompleteness Theorem




1933 Hitler comes to power.1930 Announces (First) Incompleteness Theorem




1933 Hitler comes to power.

1940 Back to USA, for good.







1978 Princeton NJ USA.







1978 Princeton NJ USA.


1936 Schlick murdered; Austria annexed

Preliminaries: Propositional Calculus &

First-Order Logic…

Actually …

ℒ0 < ℒ1 < ℒ2 < ℒ3…

Actually …

ℒ0 < ℒ1 < ℒ2 < ℒ3…

Zero-order logic; subsumes the propositional calculus. First-order logic; this is

whaat the Completeness Theorem is about.

Second-order logic. Third-order logic, which Gödel used for his “God Theorem.”

R&W’s Axiomatization of the Propositional Calculus

A1 (� _ �) ! �A2 �! (� _ )A3 (� _ ) ! ( _ �)A4 ( ! �) ! ((� _ ) ! (� _ �))

R&W’s Axiomatization of the Propositional Calculus

A1 (� _ �) ! �A2 �! (� _ )A3 (� _ ) ! ( _ �)A4 ( ! �) ! ((� _ ) ! (� _ �))

All instances of these schemata are true no matter what the input (true or false). (Agreed?) And indeed every single formula in the propositional calculus that is true no matter what the permutation (as shown in a truth table), can be proved (somehow) from these four axioms (using the rules of inference given earlier in our semester). This, Gödel knew, and could use.

Exercise 1: Verify that these are true-no-matter what in a truth table;

then prove using our rules for the prop. calc.

(� ^ ) ! ( _ �)�! ( ! �)



(� ^ ) ! ( _ �)



(� ^ ) ! ( _ �)√Truth Table showing this formula true no matter what the inputs.




Proof:




Proof:

1 � ^ Supposition2 1, Simplification3 _ � 2, Addition4 (� _ ) ! ( _ �) 1–3, Conditional Intro

From Language-Learning Slides: The Grammar of the Pure Predicate Calculus

Formula ) AtomicFormula| (Formula Connective Formula)| ¬ Formula

AtomicFormula ) (Predicate Term1 . . .Termk)| (Term = Term)

Term ) (Function Term1 . . . Termk)| Constant

Connective ) ^ | _ | ! | $

Predicate ) P1 | P2 | P3 . . .Constant ) c1 | c2 | c3 . . .Function ) f1 | f2 | f3 . . .

Recall the Examples We Cited

Sally likes Bill.(Likes sally bill)Likes

Sally likes Bill and Bill likes Sally.

Sally likes Bill’s mother.

Sally likes Bill only if Bill’s mother is tall.

5 plus 5 equals the number 10.

…




Connective ) ^ | _ | ! | $


Matilda is Bill’s super-smart mother.

Lexicon

Did you make sure you can simulate a machine that says “Yes that sentence is okay!” whenever it’s conforms to this grammar?

Recall the Examples We Cited

Sally likes Bill.(Likes sally bill)

Likes

Sally likes Bill and Bill likes Sally.

Sally likes Bill’s mother.

Sally likes Bill only if Bill’s mother is tall.

5 plus 5 equals the number 10.

…




Connective ) ^ | _ | ! | $


Matilda is Bill’s super-smart mother.

Lexicon

Did you make sure you can simulate a machine that says “Yes that sentence is okay!” whenever it’s conforms to this grammar?

And We Noted These Examples




Connective ) ^ | _ | ! | $



If Sally likes Bill then Sally likes Bill.




Connective ) ^ | _ | ! | $




Sally likes Bill’s mother, or not.




Connective ) ^ | _ | ! | $





Sally likes Bill and Bill likes Jane, only if Bill likes Jane.




Connective ) ^ | _ | ! | $









Connective ) ^ | _ | ! | $


Bill’s smart mother is a mother.





…




Connective ) ^ | _ | ! | $







…




Connective ) ^ | _ | ! | $



These are all true, yes; but can they be proved?!

Add the Final Addition/Deeper Challenge:Add Two Quantifiers to the Pure Predicate Calculus,

Which Yields the First-order Logic = Predicate Calculus simpliciter



there exists at least one thing x such that …




for all x, it’s the case that …





9x . . .





9x . . .

8x . . .





9x . . .

8x . . .

8✏(✏ > 0 ! 9�(� > 0 ^ 8x(d(x, a) < � ! d(f(x), b) < ✏)))





9x . . .

8x . . .





9x . . .

8x . . .

Every natural number is greater than or equal to zero.





9x . . .

8x . . .

Every natural number is greater than or equal to zero.8x(x � 0)





9x . . .

8x . . .


There’s a positive integer greater than any positive integer.





9x . . .

8x . . .


There’s a positive integer greater than any positive integer.9x8y(y < x)





9x . . .

8x . . .


9x8y(y < x)





9x . . .

8x . . .






9x . . .

8x . . .


Every positive integer x is less-than-or-equal-to a positive integer y.





9x . . .

8x . . .


Every positive integer x is less-than-or-equal-to a positive integer y.8x9y(x y) 8x9y( (x, y))

The Shoulders Available to Gödel for Standing Upon

…

Completeness Theorem for The Propositional Calculus

Let � be a set {�1,�2, . . .} of formulae in the the propositional calculus.Then either all of � are satisfiable, or the conjunction up to and includingthe point k (i.e. �1 ^ �2 ^ . . . ^ �k) of failure is refutable.

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Completeness Theorem for The Propositional Calculus

Let � be a set {�1,�2, . . .} of formulae in the the propositional calculus.Then either all of � are satisfiable, or the conjunction up to and includingthe point k (i.e. �1 ^ �2 ^ . . . ^ �k) of failure is refutable.

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Let � be a set {�1,�2, . . .} of formulae in the the propositional calculus.Then either all of � can be simultaneously true in some scenario, or theconjunction up to and including the point k (i.e. �1 ^ �2 ^ . . . ^ �k) offailure is refutable (i.e. ` ¬(�1 ^ �2 ^ . . . ^ �k)).

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What does the Completeness Theorem

say?…

Completeness Theorem as an Equation

In first-order logic: NECESSARY TRUTH = PROVABILITY.

Completeness Theorem, More Precisely Put

For every first-order statement : is a necessary or absolute truth (i.e. true in any scenario whatsoever) if, and only if, is provable.

ϕ ϕ

ϕ

And the version Gödel targeted, and proved:

For every first-order statement : Either is true in some scenarios, or is refutable (= it’s negation can be proved).

ϕ ϕϕ

¬ϕ

CompG

The Proof-Sketch

The Proof-SketchTo prove the theorem in the case of first-order logic , we need to show that given any set of formulae in first-order logic, either there's a scenario on which every member of this set is true; otherwise, there is a refutation of the set, i.e. a proof from the set to an outright contradiction . We can accomplish this by finding a procedure that first takes the set in question and goes hunting for a scenario that does the trick. If the scenario is found, we're done. But, if such a scenario can't be found, then our procedure moves on to find a proof of a contradiction from !

( = ℒ1)Γ

ϕ ∧ ¬ϕ𝒫

Γ

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X X¬p2

<latexit sha1_base64="8akOZNoW0cE8i2IcwzZ0Ry/Nd7A=">AAACM3icbVBLSwMxGEzqq9ZXq3jyEiyCp7JbBT0WvXisYB/QlpLNZtvQbLIk3wpl6Z/wqn/DHyPexKv/wfRxsK0DgWG+GZhMkEhhwfM+cG5jc2t7J79b2Ns/ODwqlo6bVqeG8QbTUpt2QC2XQvEGCJC8nRhO40DyVjC6n95bz9xYodUTjBPei+lAiUgwCk5qdxUfkKRf7RfLXsWbgawTf0HKaIF6v4RPu6FmacwVMEmt7fheAr2MGhBM8kmhm1qeUDaiA95xVNGY2142KzwhF04JSaSNewrITP2byGhs7TgOnDOmMLRLNwsxNWMTTlYTU+u/ialibGQnaypoLe1KWYhue5lQSQpcsXnXKJUENJkOSEJhOAM5doQyI9x3CRtSQxm4mQtuSH91tnXSrFb8q4r/eF2u3S0mzaMzdI4ukY9uUA09oDpqIIYkekGv6A2/40/8hb/n1hxeZE7QEvDPLwgwqzs=</latexit>

p4 ! p1<latexit sha1_base64="H+BljCbuV4tbOyQ6UUTuJ4LI/0w=">AAACQnicbVBLS8NAGNzUV62vVunJy2IRPJVEC3osevFYwT6gKWGz2bRLN9mw+0UpIT/Gq/4N/4R/wZt49eD2cbCtAwvDfDMwO34iuAbb/rAKG5tb2zvF3dLe/sHhUbly3NEyVZS1qRRS9XyimeAxawMHwXqJYiTyBev647vpvfvElOYyfoRJwgYRGcY85JSAkbxyNfEaLnYVH46AKCWfXZx4jleu2XV7BrxOnAWpoQVaXsWquoGkacRioIJo3XfsBAYZUcCpYHnJTTVLCB2TIesbGpOI6UE265/jc6MEOJTKvBjwTP2byEik9STyjTMiMNJLNw0RURMV5KuJqfXfxFRROtT5mgpSCr1SFsKbQcbjJAUW03nXMBUYJJ7uiQOuGAUxMYRQxc13MR0RRSiY1UtmSGd1tnXSuaw7V3XnoVFr3i4mLaJTdIYukIOuURPdoxZqI4oy9IJe0Zv1bn1aX9b33FqwFpkTtATr5xdvA7DX</latexit>

p4 true; p1 true<latexit sha1_base64="EJKvWy8YP88y94/fnZI7sfGhfSg=">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</latexit>

p4 false; p1 false<latexit sha1_base64="LduelHnvEsLrFBRp/zmCnF9q4NM=">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</latexit>

X Xp4 false; p1 true

<latexit sha1_base64="gTLDHBcjVme+PH2zneOXjSz2ZFY=">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</latexit>

X

� := {p1 ! p2, p3 ^ p4, ¬p2, p4 ! p1, . . .}<latexit sha1_base64="QZ4q5sNsriK7c63W6kDlO730+JM=">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</latexit>

Therefore, there is no scenario in which all of the formulae are true!

Therefore, since we can travel to infinity, there is a scenario in which all of the formulae are true: any infinite path down will do.

� := {p1 ! p3, p2, p4 ! p6, p5, p7 ! p9, p8, . . .}<latexit sha1_base64="o3WX/XN2n7CzNpWmLuNt7gNiu5Y=">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</latexit>

p1 ! p3<latexit sha1_base64="VPCC9++h6xkj6sbBM72+oMrk+8s=">AAACQHicbVDNSsNAGNzUv1r/WosnL4tF8FQSK+ix6MVjBfsDbQibzaZdusmG3S9KKH0Wr/oavoVv4E28enLT9mBbBxaG+WZgdvxEcA22/WEVNja3tneKu6W9/YPDo3LluKNlqihrUymk6vlEM8Fj1gYOgvUSxUjkC9b1x3f5vfvElOYyfoQsYW5EhjEPOSVgJK9cTTwHDxQfjoAoJZ9x4jW8cs2u2zPgdeIsSA0t0PIq1skgkDSNWAxUEK37jp2AOyEKOBVsWhqkmiWEjsmQ9Q2NScS0O5m1n+JzowQ4lMq8GPBM/ZuYkEjrLPKNMyIw0ks3DRFRmQqmq4nc+m8iV5QO9XRNBSmFXikL4Y074XGSAovpvGuYCgwS52vigCtGQWSGEKq4+S6mI6IIBbN5yQzprM62TjqXdadRdx6uas3bxaRFdIrO0AVy0DVqonvUQm1EUYZe0Ct6s96tT+vL+p5bC9YiU0VLsH5+Acv+sAo=</latexit>

p1 true; p3 true<latexit sha1_base64="QuoZlSQIb6PHWWPhLzR7vFedX14=">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</latexit>

p1 false; p3 false<latexit sha1_base64="NXHlJWyy+sHzpYhE1ZzfgqtLbac=">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</latexit>

p1 false; p3 true<latexit sha1_base64="3MLQBKFXu9f0lWg6eVB5Jpcit2k=">AAACY3icbVHLSiNBFK20OmqcGeNrJUIxQZhV6FZRwY3oxqWCUcEOTXXlthZWdTV1b4uh6T/xa9zqD/gB8x9TiVlo4oGCwzn3wL2n0kIrpDB8bwQzs3M/5hcWm0s/f/1ebq2sXqEtnYSutNq6m1QgaJVDlxRpuCkcCJNquE4fTof+9SM4VDa/pEEBPSPucpUpKchLSWu/SKKYxya1T1VM8EQoq0xohLo+inmR7E6a5ErvJa122AlH4NMkGpM2G+M8WWlsxH0rSwM5SS0Qb6OwoF4lHCmpoW7GJUIh5IO4g1tPc2EAe9XowJpve6XPM+v8y4mP1M+JShjEgUn9pBF0j188JCPcwPXrycRw9NvEUHGYYT2lkrUaJ5al7LBXqbwoCXL5sWtWak6WDwvnfeVAkh54IqRT/lwu74UTkvy3NH2R0WRt0+RqpxPtdqKLvfbxybjSBbbJ/rC/LGIH7JidsXPWZZI9sxf2yt4a/4KlYDVY/xgNGuPMGvuCYOs/T5278w==</latexit>

p2<latexit sha1_base64="86N1taBFDKSWniqV7ZgEdadVCyo=">AAACLnicbVDNSsNAGNz4W+tfq3jyEiyCp5JUQY9FLx4r2h9oQ9lsNu3SzW7Y/SKEkkfwqq/h0wgexKuP4abNwbYOLAzzzcDs+DFnGhzn01pb39jc2i7tlHf39g8OK9WjjpaJIrRNJJeq52NNORO0DQw47cWK4sjntOtP7vJ795kqzaR4gjSmXoRHgoWMYDDSYzxsDCs1p+7MYK8StyA1VKA1rFong0CSJKICCMda910nBm+KFTDCaVYeJJrGmEzwiPYNFTii2pvOumb2uVECO5TKPAH2TP2bmOJI6zTyjTPCMNYLNw0RVqkKsuVEbv03kStKhzpbUUFKrpfKQnjjTZmIE6CCzLuGCbdB2vl2dsAUJcBTQzBRzHzXJmOsMAGzcNkM6S7Ptko6jbp7WXcfrmrN22LSEjpFZ+gCuegaNdE9aqE2ImiEXtArerPerQ/ry/qeW9esInOMFmD9/AI4x6lT</latexit>

p4 ! p6<latexit sha1_base64="rvV3KSXtypc/SM3s2JmeCKmLOTg=">AAACQHicbVBLS8NAGNz4rPXVWjx5WSyCp5JoUY9FLx4r2Ae0IWw2m3bpJht2vyih5Ld41b/hv/AfeBOvntw+DrZ1YGGYbwZmx08E12DbH9ba+sbm1nZhp7i7t39wWCoftbVMFWUtKoVUXZ9oJnjMWsBBsG6iGIl8wTr+6G5y7zwxpbmMHyFLmBuRQcxDTgkYyStVEq+O+4oPhkCUks848a68UtWu2VPgVeLMSRXN0fTK1nE/kDSNWAxUEK17jp2AOyYKOBUsL/ZTzRJCR2TAeobGJGLaHU/b5/jMKAEOpTIvBjxV/ybGJNI6i3zjjAgM9cJNQ0RUpoJ8OTGx/puYKEqHOl9RQUqhl8pCeOOOeZykwGI66xqmAoPEkzVxwBWjIDJDCFXcfBfTIVGEgtm8aIZ0lmdbJe2LmnNZcx7q1cbtfNICOkGn6Bw56Bo10D1qohaiKEMv6BW9We/Wp/Vlfc+sa9Y8U0ELsH5+AdbesBA=</latexit>

p4 true; p6 true<latexit sha1_base64="NedoU/0g7uIvVhvnrqrotOuceSA=">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</latexit>

p4 false; p6 false<latexit sha1_base64="O3s5HlOBPu9WSfdS0cAfTArREUU=">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</latexit>

p4 false; p6 true<latexit sha1_base64="y15wsPDpkNdSgUTDnEwXwx3VFJc=">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</latexit>

p2 true<latexit sha1_base64="0VeFMQgaXI9uQSLSTJWi+dp6X/s=">AAACRnicbVDLahtBEOyV81CUl+SQQ8hliAjkJHblQHIU9iVHBSJLoBVidtQrDZ7ZWWZ6g8Sy+Gt8tX/Dv+CfyC346pGsQySloKGoroLuSnIlHYXhXVA7evL02fP6i8bLV6/fvG22js+dKazAgTDK2FHCHSqZ4YAkKRzlFrlOFA6Ti7P1fvgbrZMm+0WrHCeazzOZSsHJS9Pmh3zajVmsE7MsY8IlOVGSLbCqps122Ak3YIck2pI2bNGftoL38cyIQmNGQnHnxlGY06TklqRQWDXiwmHOxQWf49jTjGt0k3LzQ8U+e2XGUmP9ZMQ26r+JkmvnVjrxTs1p4XZ2jjS3Kzur9hNr638Ta8W61FUHKhmj3N6xlH6flDLLC8JMPN6aFoqRYetO2UxaFKRWnnBhpX+XiQW3XJBvvuGLjPZrOyTn3U500ol+fm33TreV1uEjfIIvEME36MEP6MMABFzCFVzDTXAb/An+BveP1lqwzbyDHdTgAcUGspY=</latexit>









p5<latexit sha1_base64="zQBKeE+dacApeLKvft6OfbOe7FQ=">AAACLnicbVBLS8NAGNz4rPXVKp68LBbBU0l8oMeiF48V7QPaUDabTbt0kw27X4RQ8hO86t/w1wgexKs/w02bg20dWBjmm4HZ8WLBNdj2p7Wyura+sVnaKm/v7O7tV6oHbS0TRVmLSiFV1yOaCR6xFnAQrBsrRkJPsI43vsvvnWemNJfRE6Qxc0MyjHjAKQEjPcaDq0GlZtftKfAycQpSQwWag6p11PclTUIWARVE655jx+BOiAJOBcvK/USzmNAxGbKeoREJmXYn064ZPjWKjwOpzIsAT9W/iQkJtU5DzzhDAiM9d9MQEpUqP1tM5NZ/E7midKCzJRWkFHqhLAQ37oRHcQIsorOuQSIwSJxvh32uGAWRGkKo4ua7mI6IIhTMwmUzpLM42zJpn9edi7rzcFlr3BaTltAxOkFnyEHXqIHuURO1EEVD9IJe0Zv1bn1YX9b3zLpiFZlDNAfr5xc+H6lW</latexit>

p5 true<latexit sha1_base64="JvTnUEVxEj792Nvr/NVgyi0Yorc=">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</latexit>









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But the assumption that there is an infinite branch is based on König’s Lemma …

Toward König’s Lemma as Train Travel

“To infinity and beyond!”

König’s Lemma (train-travel version)

In a one-way train-travel map with finitely many options leading from each station, if there are partial paths forward of every finite length, there is an infinite path (= a path “to infinity”).

Exercise 2: Is there an algorithm for traveling this way?


No. This strategy for travel is beyond the reach of standard computation.


No. This strategy for travel is beyond the reach of standard computation.

(Does it not then follow, assuming that humans can find and “use” a provably correct strategy for this travel, that humans can’t be fundamentally computing machines?)

Proving the Lemma(that there is an infinite branch)

Proof: We are seeking to prove that there is an infinite path (= that you can keep going forward forever = that the number of your stops forward are the size of Z+).

To begin, assume the antecedent of the theorem (i.e. that, (1), there are finitely many options leading from each station, and that, (2), in the map there are partial paths forward of every finite size).

Now, you are standing at Penn Station (S1), facing k options. At least one of these options must lead to partial paths of arbitrary size (the size of any m in Z+). (Sub-Proof: Suppose otherwise for indirect proof. Then there is some positive integer n that places a ceiling on the size of partial paths that can be reached. But this violates (2) — contradiction.) Proceed to choose one of these options that lead to partial paths of arbitrary size. You are now standing at a new station (S2), one stop after Penn Station. At least one of these options must lead to partial parts of arbitrary size (the size of any m in Z+). (Sub-Proof: Suppose otherwise for indirect proof …)

Since you can iterate this forever, you’ll be on an infinite trip to infinity! Buzz will be happy.

For Further Reading

slutten

Govindarajulu, N.; Licato, J.; Bringsjord, S. 2013. Small Steps Toward Hypercomputation via Infinitary Machine Proof Verification and Proof Generation. In Proceedings of UCNC 2013. Pdf

http://link.springer.com/chapter/10.1007/978-3-642-39074-6_11

Needs Understanding of Ordinal Numbers …

191 = 2222

0

+ 220

+ 20 < !!!!0

+ !!0

+ !0


191 = 2222

0

+ 220

+ 20 < !!!!0

+ !!0

+ !0

192 = 3333

0

+ 330

+ 30 � 1 < !!!!0

+ !!0

+ !0 � 1


191 = 2222

0

+ 220

+ 20 < !!!!0

+ !!0

+ !0

192 = 3333

0

+ 330

+ 30 � 1 < !!!!0

+ !!0

+ !0 � 1

193 = 4444

0

+ 440

� 1 < !!!!0

+ !!0

� 1


191 = 2222

0

+ 220

+ 20 < !!!!0

+ !!0

+ !0

192 = 3333

0

+ 330

+ 30 � 1 < !!!!0

+ !!0

+ !0 � 1

193 = 4444

0

+ 440

� 1 < !!!!0

+ !!0

� 1

194 = 5555

0

+ 50 + 50 � 1 < !!!!0

+ !0 + !0 � 1


191 = 2222

0

+ 220

+ 20 < !!!!0

+ !!0

+ !0

192 = 3333

0

+ 330

+ 30 � 1 < !!!!0

+ !!0

+ !0 � 1

193 = 4444

0

+ 440

� 1 < !!!!0

+ !!0

� 1

194 = 5555

0

+ 50 + 50 � 1 < !!!!0

+ !0 + !0 � 1

195 = 6666

0

+ 60 < !!!!0

+ !0


191 = 2222

0

+ 220

+ 20 < !!!!0

+ !!0

+ !0

192 = 3333

0

+ 330

+ 30 � 1 < !!!!0

+ !!0

+ !0 � 1

193 = 4444

0

+ 440

� 1 < !!!!0

+ !!0

� 1

194 = 5555

0

+ 50 + 50 � 1 < !!!!0

+ !0 + !0 � 1

195 = 6666

0

+ 60 < !!!!0

+ !0

...


191 = 2222

0

+ 220

+ 20 < !!!!0

+ !!0

+ !0

192 = 3333

0

+ 330

+ 30 � 1 < !!!!0

+ !!0

+ !0 � 1

193 = 4444

0

+ 440

� 1 < !!!!0

+ !!0

� 1

194 = 5555

0

+ 50 + 50 � 1 < !!!!0

+ !0 + !0 � 1

195 = 6666

0

+ 60 < !!!!0

+ !0

...

strictly decreasing


Ordinal Numbers …

Yet, Conjecture (C) (see “Isaacson’s Conjecture”)


In order to produce a rationally compelling proof of any true sentence S formed from the symbol set of the language of arithmetic, but independent of PA, it’s necessary to deploy concepts and structures of an irreducibly infinitary nature.


In order to produce a rationally compelling proof of any true sentence S formed from the symbol set of the language of arithmetic, but independent of PA, it’s necessary to deploy concepts and structures of an irreducibly infinitary nature.

If this is right, and computing machines can’t use irreducibly infinitary techniques, they’re in trouble — or: there won’t be a Singularity.

gödel’s completeness theoremkryten.mm.rpi.edu/courses/ahr/godel_completeness_part2.pdf ·...

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