steeple #2 - rensselaer polytechnic...
TRANSCRIPT
![Page 1: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/1.jpg)
Steeple #2:Gödel’s First Incompleteness Theorem
(The “Parlor Trick” Theorem)(and thereafter: Steeple #3: The “Silver Blaze” (from Sherlock Holmes) Theorem)
Selmer BringsjordAre Humans Rational?
Dec 5 2019RPI
Troy NY USA
![Page 2: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/2.jpg)
Background Context …
![Page 3: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/3.jpg)
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?
![Page 4: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/4.jpg)
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?
![Page 5: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/5.jpg)
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?
A corollary of the First Incompleteness Theorem: We cannot prove that mathematics is consistent.
![Page 6: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/6.jpg)
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?
![Page 7: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/7.jpg)
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?
Based on the Sherlock Holmes mystery “Silver Blaze”; read for next (& last) class).
![Page 8: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/8.jpg)
Some Timeline Points
1906 Brünn, Austria-Hungary1923 Vienna
Undergrad in seminar by Schlick1929 Doctoral Dissertation: Proof of Completeness Theorem
1933 Hitler comes to power.
1940 Back to USA, for good.
1978 Princeton NJ USA.
1930 Announces (First) Incompleteness Theorem
1936 Schlick murdered; Austria annexed
![Page 9: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/9.jpg)
Some Timeline Points
1906 Brünn, Austria-Hungary1923 Vienna
Undergrad in seminar by Schlick1929 Doctoral Dissertation: Proof of Completeness Theorem
1933 Hitler comes to power.
1940 Back to USA, for good.
1978 Princeton NJ USA.
1930 Announces (First) Incompleteness Theorem
1936 Schlick murdered; Austria annexed
![Page 10: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/10.jpg)
Some Timeline Points
1906 Brünn, Austria-Hungary1923 Vienna
Undergrad in seminar by Schlick1929 Doctoral Dissertation: Proof of Completeness Theorem
1933 Hitler comes to power.
1940 Back to USA, for good.
1978 Princeton NJ USA.
1930 Announces (First) Incompleteness Theorem
1936 Schlick murdered; Austria annexed
![Page 11: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/11.jpg)
Some Timeline Points
1906 Brünn, Austria-Hungary1923 Vienna
Undergrad in seminar by Schlick1929 Doctoral Dissertation: Proof of Completeness Theorem
1933 Hitler comes to power.
1940 Back to USA, for good.
1978 Princeton NJ USA.
1930 Announces (First) Incompleteness Theorem
1936 Schlick murdered; Austria annexed
![Page 12: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/12.jpg)
Steeple #2 (Incompleteness) …
![Page 13: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/13.jpg)
Goal: Put you in position to prove Gödel’s first incompleteness theorem!
![Page 14: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/14.jpg)
Goal: Put you in position to prove Gödel’s first incompleteness theorem!
We have the background (if).
![Page 15: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/15.jpg)
The “Liar Tree”
![Page 16: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/16.jpg)
The “Liar Tree”The Liar Paradox
![Page 17: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/17.jpg)
The “Liar Tree”The Liar Paradox
![Page 18: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/18.jpg)
The “Liar Tree”The Liar Paradox
Pure Proof-Theoretic Route
![Page 19: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/19.jpg)
The “Liar Tree”The Liar Paradox
Pure Proof-Theoretic Route
![Page 20: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/20.jpg)
The “Liar Tree”The Liar Paradox
Via “Tarski’s” TheoremPure Proof-Theoretic Route
![Page 21: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/21.jpg)
The “Liar Tree”The Liar Paradox
Via “Tarski’s” TheoremPure Proof-Theoretic Route
![Page 22: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/22.jpg)
The “Liar Tree”The Liar Paradox
Via “Tarski’s” TheoremPure Proof-Theoretic Route
![Page 23: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/23.jpg)
The “Liar Tree”The Liar Paradox
Via “Tarski’s” TheoremPure Proof-Theoretic Route
Ergo, step one: What is LP?
![Page 24: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/24.jpg)
Remember …“The (Economical) Liar” …
![Page 25: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/25.jpg)
Remember …“The (Economical) Liar” …
L: This sentence is false.
![Page 26: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/26.jpg)
Remember …“The (Economical) Liar” …
L: This sentence is false.
Suppose that T(L); then ¬T(L).
![Page 27: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/27.jpg)
Remember …“The (Economical) Liar” …
L: This sentence is false.
Suppose that T(L); then ¬T(L).
Suppose that ¬T(L) then T(L).
![Page 28: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/28.jpg)
Remember …“The (Economical) Liar” …
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).
![Page 29: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/29.jpg)
Remember …“The (Economical) Liar” …
L: This sentence is false.
Suppose that T(L); then ¬T(L).
Suppose that ¬T(L) then T(L).
Contradiction!
Hence: T(L) iff (i.e., if & only if) ¬T(L).
![Page 30: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/30.jpg)
The “Gödelian” Liar
![Page 31: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/31.jpg)
The “Gödelian” Liar: This sentence is unprovable.P̄
![Page 32: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/32.jpg)
The “Gödelian” Liar: This sentence is unprovable.P̄
Suppose that is true. Then we can immediately deduce that is provable, because here is a proof: is an easy theorem, and from it and our supposition we deduce by modus ponens. But since what says is that it’s unprovable, we have deduced that is false under our initial supposition.
P̄P̄ P̄ → P̄
P̄P̄
P̄
![Page 33: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/33.jpg)
The “Gödelian” Liar: This sentence is unprovable.P̄
Suppose that is true. Then we can immediately deduce that is provable, because here is a proof: is an easy theorem, and from it and our supposition we deduce by modus ponens. But since what says is that it’s unprovable, we have deduced that is false under our initial supposition.
P̄P̄ P̄ → P̄
P̄P̄
P̄
Suppose on the other hand that is false. Then we can immediately deduce that is unprovable: Suppose for reductio that
is provable; then holds as a result of some proof, but what says is that it’s unprovable; and so we have contradiction. But since what says is that it’s unprovable, and we have just proved that under our supposition, we arrive at the conclusion that is true.
P̄P̄
P̄ P̄ P̄
P̄P̄
![Page 34: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/34.jpg)
The “Gödelian” Liar: This sentence is unprovable.P̄
Suppose that is true. Then we can immediately deduce that is provable, because here is a proof: is an easy theorem, and from it and our supposition we deduce by modus ponens. But since what says is that it’s unprovable, we have deduced that is false under our initial supposition.
P̄P̄ P̄ → P̄
P̄P̄
P̄
T( ) iff (i.e., if & only if) ¬T( ) = F( )P̄ P̄ P̄
Suppose on the other hand that is false. Then we can immediately deduce that is unprovable: Suppose for reductio that
is provable; then holds as a result of some proof, but what says is that it’s unprovable; and so we have contradiction. But since what says is that it’s unprovable, and we have just proved that under our supposition, we arrive at the conclusion that is true.
P̄P̄
P̄ P̄ P̄
P̄P̄
![Page 35: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/35.jpg)
The “Gödelian” Liar: This sentence is unprovable.P̄
Suppose that is true. Then we can immediately deduce that is provable, because here is a proof: is an easy theorem, and from it and our supposition we deduce by modus ponens. But since what says is that it’s unprovable, we have deduced that is false under our initial supposition.
P̄P̄ P̄ → P̄
P̄P̄
P̄
Contradiction!T( ) iff (i.e., if & only if) ¬T( ) = F( )P̄ P̄ P̄
Suppose on the other hand that is false. Then we can immediately deduce that is unprovable: Suppose for reductio that
is provable; then holds as a result of some proof, but what says is that it’s unprovable; and so we have contradiction. But since what says is that it’s unprovable, and we have just proved that under our supposition, we arrive at the conclusion that is true.
P̄P̄
P̄ P̄ P̄
P̄P̄
![Page 36: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/36.jpg)
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
where �(x) is open w� with variable x, and perhaps others, free.([�(0) ⇤ ⇥x(�(x) � �(s(x))] � ⇥x�(x))
![Page 37: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/37.jpg)
Arithmetic is Part of All Things Sci/Eng/Tech!and courtesy of Gödel: We can’t even prove all truths of arithmetic!
…
PA … …
Each circle is a larger part of the formal sciences.
![Page 38: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/38.jpg)
Gödel Numbering
![Page 39: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/39.jpg)
Gödel NumberingProblem: How do enables a formula to refer to other formula and itself (and also other objects like proofs, terms etc.), in a perfectly consistent way?
![Page 40: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/40.jpg)
Gödel NumberingProblem: How do enables a formula to refer to other formula and itself (and also other objects like proofs, terms etc.), in a perfectly consistent way?
Solution: Gödel numbering
![Page 41: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/41.jpg)
Gödel Numbering
�
�!
f(x, a)
Problem: How do enables a formula to refer to other formula and itself (and also other objects like proofs, terms etc.), in a perfectly consistent way?
Solution: Gödel numbering
![Page 42: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/42.jpg)
Gödel Numbering
Syntactic objects
�
�!
f(x, a)
Problem: How do enables a formula to refer to other formula and itself (and also other objects like proofs, terms etc.), in a perfectly consistent way?
Solution: Gödel numbering
![Page 43: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/43.jpg)
Gödel Numbering
Syntactic objects
�
�!
f(x, a)
Problem: How do enables a formula to refer to other formula and itself (and also other objects like proofs, terms etc.), in a perfectly consistent way?
Solution: Gödel numbering
(formulae, terms, proofs etc)
![Page 44: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/44.jpg)
Gödel Numbering
Syntactic objects
�
�!
f(x, a)
Problem: How do enables a formula to refer to other formula and itself (and also other objects like proofs, terms etc.), in a perfectly consistent way?
Solution: Gödel numbering
(formulae, terms, proofs etc)
![Page 45: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/45.jpg)
Gödel Numbering
Syntactic objects
�
�!
f(x, a)
Problem: How do enables a formula to refer to other formula and itself (and also other objects like proofs, terms etc.), in a perfectly consistent way?
Solution: Gödel numbering
(formulae, terms, proofs etc)
323
23432
142323
![Page 46: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/46.jpg)
Gödel Numbering
Gödel numberSyntactic objects
�
�!
f(x, a)
Problem: How do enables a formula to refer to other formula and itself (and also other objects like proofs, terms etc.), in a perfectly consistent way?
Solution: Gödel numbering
(formulae, terms, proofs etc)
323
23432
142323
![Page 47: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/47.jpg)
Gödel Numbering
Gödel numberSyntactic objects
�
�!
f(x, a)
Problem: How do enables a formula to refer to other formula and itself (and also other objects like proofs, terms etc.), in a perfectly consistent way?
Solution: Gödel numbering
(formulae, terms, proofs etc)
323
23432
142323
![Page 48: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/48.jpg)
Gödel Numbering
Gödel numberSyntactic objects
�
�!
f(x, a)
Problem: How do enables a formula to refer to other formula and itself (and also other objects like proofs, terms etc.), in a perfectly consistent way?
Solution: Gödel numbering
(formulae, terms, proofs etc)
323
23432
142323
1+...+1+0
1+...+1+0
1+...+1+0
![Page 49: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/49.jpg)
Gödel Numbering
Gödel number Gödel numeralSyntactic objects
�
�!
f(x, a)
Problem: How do enables a formula to refer to other formula and itself (and also other objects like proofs, terms etc.), in a perfectly consistent way?
Solution: Gödel numbering
(formulae, terms, proofs etc)
323
23432
142323
1+...+1+0
1+...+1+0
1+...+1+0
![Page 50: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/50.jpg)
Gödel Numbering
Gödel number Gödel numeralSyntactic objects
�
�!
f(x, a)
Problem: How do enables a formula to refer to other formula and itself (and also other objects like proofs, terms etc.), in a perfectly consistent way?
Solution: Gödel numbering
(formulae, terms, proofs etc)
323
23432
142323
back to syntax
1+...+1+0
1+...+1+0
1+...+1+0
![Page 51: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/51.jpg)
Gödel Numbering
Gödel number Gödel numeralSyntactic objects
�
�!
f(x, a)
Problem: How do enables a formula to refer to other formula and itself (and also other objects like proofs, terms etc.), in a perfectly consistent way?
Solution: Gödel numbering
(formulae, terms, proofs etc)
323
23432
142323
back to syntax
1+...+1+0
1+...+1+0
1+...+1+0
�
![Page 52: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/52.jpg)
Gödel Numbering
Gödel number Gödel numeralSyntactic objects
�
�!
f(x, a)
Problem: How do enables a formula to refer to other formula and itself (and also other objects like proofs, terms etc.), in a perfectly consistent way?
Solution: Gödel numbering
(formulae, terms, proofs etc)
323
23432
142323
back to syntax
1+...+1+0
1+...+1+0
1+...+1+0
� n�
![Page 53: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/53.jpg)
Gödel Numbering
Gödel number Gödel numeralSyntactic objects
�
�!
f(x, a)
Problem: How do enables a formula to refer to other formula and itself (and also other objects like proofs, terms etc.), in a perfectly consistent way?
Solution: Gödel numbering
(formulae, terms, proofs etc)
323
23432
142323
back to syntax
1+...+1+0
1+...+1+0
1+...+1+0
� n� n̂� (or just )“�”
![Page 54: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/54.jpg)
Gödel Numbering, the Easy Way
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 …
![Page 55: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/55.jpg)
Gödel Numbering, the Easy Way
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”.
![Page 56: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/56.jpg)
Gödel Numbering, the Easy Way
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 …
![Page 57: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/57.jpg)
Gödel Numbering, the Easy Way
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 …
Or, every syntactically valid computer program in Haskell that you will ever write can be uniquely picked about some number m in the lexicographic ordering of all syntactically valid such programs, and then we can just use a numeral or string “m” (or whatever) to refer to your program in a formal language/logic!
![Page 58: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/58.jpg)
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*)!
Gödel’s First Incompleteness Theorem
![Page 59: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/59.jpg)
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*)!
Gödel’s First Incompleteness Theorem
(Oh, and: g* is true!)
![Page 60: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/60.jpg)
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 2 + 3 = 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”)
(i.e., a formula g that says: “I have property q!”)
Proof Kernel for Theorem GIPart 1: Recipe R
![Page 61: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/61.jpg)
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)
Proof Kernel for Theorem GIPart 2: Follow Recipe R, Guided by The Liar
Next, we follow Gödel’s Recipe R to build a g* such that:
PA |- g* <=> not-q*(“g*”) (2)
![Page 62: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/62.jpg)
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)
Proof Kernel for Theorem GIPart 2: Follow Recipe R, Guided by The Liar
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*?!?
![Page 63: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/63.jpg)
PA |- q*(“g*”) iff PA |- g* (1’)
Indirect Proof
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:
GI: g* isn’t provable from PA; nor is the negation of g*!
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. But we are working under the supposition that PA is consistent. Hence by indirect proof g* is not provable from PA.
![Page 64: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/64.jpg)
PA |- q*(“g*”) iff PA |- g* (1’)
Indirect Proof
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:
GI: g* isn’t provable from PA; nor is the negation of g*!
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. But we are working under the supposition that PA is consistent. 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?
![Page 65: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/65.jpg)
“Silly abstract nonsense! There aren’t any concrete examples of g*!”
![Page 66: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/66.jpg)
Ah, but: Goodstein’s Theorem!
![Page 67: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/67.jpg)
Ah, but: Goodstein’s Theorem!
The Goodstein Sequence goes to zero!
![Page 68: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/68.jpg)
Pure base n representation of a number r
• Represent r as only sum of powers of n in which the exponents are also powers of n etc
![Page 69: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/69.jpg)
Grow Function
![Page 70: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/70.jpg)
Example of Grow
![Page 71: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/71.jpg)
Goodstein Sequence• For any natural number m
![Page 72: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/72.jpg)
Sample Values
![Page 73: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/73.jpg)
Sample Values
![Page 74: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/74.jpg)
Sample Valuesm
![Page 75: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/75.jpg)
Sample Valuesm
2 2 2 1 0
![Page 76: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/76.jpg)
Sample Valuesm
2 2 2 1 0
3 3 3 3 2 1 0
![Page 77: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/77.jpg)
Sample Valuesm
2 2 2 1 0
3 3 3 3 2 1 0
4 4 26 41 60 83 109 139 ... 11327 (96th term)
...
![Page 78: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/78.jpg)
Sample Valuesm
2 2 2 1 0
3 3 3 3 2 1 0
4 4 26 41 60 83 109 139 ... 11327 (96th term)
...
5 15 ~1013 ~10155 ~102185 ~1036306 10695975 1015151337 ...
![Page 79: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/79.jpg)
Ah, but: Goodstein’s Theorem!
![Page 80: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/80.jpg)
Ah, but: Goodstein’s Theorem!
This sequence actually goes to zero!
![Page 81: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/81.jpg)
EA/BA`
EA/BA|=
Q`
Q|=
TRUEA/QTRUEA/BA
√√
PA`
PA|=
ACA0√
TRUEA/I
? G
?
…
? GT
PA`II
PA|=II= TRUEA/II
?
Astrologic: Rational Aliens Will be on the Same “Race Track”!
![Page 82: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/82.jpg)
EA/BA`
EA/BA|=
Q`
Q|=
TRUEA/QTRUEA/BA
√√
PA`
PA|=
ACA0√
TRUEA/I
? G
?
…
? GT
PA`II
PA|=II= TRUEA/II
?
Astrologic: Rational Aliens Will be on the Same “Race Track”!
![Page 83: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/83.jpg)
Could an AI Ever Match Gödel here?
![Page 84: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/84.jpg)
Actually, thanks to AFOSR:GI (& GT): “Done” ...
![Page 85: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/85.jpg)
Licato, J.; Govindarajulu, N.; Bringsjord, S.; Pomeranz, M.; Gittelson, L. 2013. Analogico-Deductive Generation of Godel’s First Incompleteness Theorem from the Liar Paradox.
In Proceedings of IJCAI 2013. PdfGovindarajulu, N.; Licato, J.; Bringsjord, S. 2013. Small Steps Toward Hypercomputation via
Infinitary Machine Proof Verification and Proof Generation. In Proceedings of UCNC 2013. Pdf
![Page 86: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/86.jpg)
Licato, J.; Govindarajulu, N.; Bringsjord, S.; Pomeranz, M.; Gittelson, L. 2013. Analogico-Deductive Generation of Godel’s First Incompleteness Theorem from the Liar Paradox.
In Proceedings of IJCAI 2013. PdfGovindarajulu, N.; Licato, J.; Bringsjord, S. 2013. Small Steps Toward Hypercomputation via
Infinitary Machine Proof Verification and Proof Generation. In Proceedings of UCNC 2013. Pdf
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Liar Paradox (L)• Collection of semiformal statements• Syntax not rigorously defined• Represents intuitive understanding of problem
domain
Continuum of Results
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)
...
![Page 88: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/88.jpg)
Liar Paradox (L)• Collection of semiformal statements• Syntax not rigorously defined• Represents intuitive understanding of problem
domain
Continuum of Results
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)
...
![Page 89: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/89.jpg)
Liar Paradox (L)• Collection of semiformal statements• Syntax not rigorously defined• Represents intuitive understanding of problem
domain
Continuum of Results
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)
...
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
...
![Page 90: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/90.jpg)
Liar Paradox (L)• Collection of semiformal statements• Syntax not rigorously defined• Represents intuitive understanding of problem
domain
Continuum of Results
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)
...
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
...
![Page 91: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/91.jpg)
Liar Paradox (L)• Collection of semiformal statements• Syntax not rigorously defined• Represents intuitive understanding of problem
domain
Continuum of Results
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)
...
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
...
![Page 92: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/92.jpg)
Liar Paradox (L)• Collection of semiformal statements• Syntax not rigorously defined• Represents intuitive understanding of problem
domain
Continuum of Results
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)
...
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
...
“Done”
![Page 93: Steeple #2 - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/COURSES/AHR/Godel_Incompleteness.pdf · 2019-12-05 · The “Gödelian” Liar P¯: This sentence is unprovable. Suppose](https://reader035.vdocument.in/reader035/viewer/2022070709/5ebef8b0607646403a7dad08/html5/thumbnails/93.jpg)
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slutten