Post on 10-Feb-2020
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?
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?
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?
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
Some Timeline Points
1906 Brünn, Austria-Hungary
Some Timeline Points
1906 Brünn, Austria-Hungary1923 Vienna
Some Timeline Points
1906 Brünn, Austria-Hungary1923 Vienna
Undergrad in seminar by Schlick
Some Timeline Points
1906 Brünn, Austria-Hungary1923 Vienna
Undergrad in seminar by Schlick1929 Doctoral Dissertation: Proof of Completeness Theorem
Some Timeline Points
1906 Brünn, Austria-Hungary1923 Vienna
Undergrad in seminar by Schlick1929 Doctoral Dissertation: Proof of Completeness Theorem
1930 Announces (First) Incompleteness Theorem
Some Timeline Points
1906 Brünn, Austria-Hungary1923 Vienna
Undergrad in seminar by Schlick1929 Doctoral Dissertation: Proof of Completeness Theorem
1933 Hitler comes to power.1930 Announces (First) Incompleteness Theorem
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.
1930 Announces (First) Incompleteness Theorem
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.
1930 Announces (First) Incompleteness Theorem
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
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
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
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.
(� ^ ) ! ( _ �)�! ( ! �)
Exercise 1: Verify that these are true-no-matter what in a truth table;
then prove using our rules for the prop. calc.
(� ^ ) ! ( _ �)
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.
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:
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:
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.
…
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 . . .
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.
…
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 . . .
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
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 . . .
And We Noted These Examples
If Sally likes Bill then Sally likes Bill.
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 . . .
And We Noted These Examples
If Sally likes Bill then Sally likes Bill.
Sally likes Bill’s mother, or not.
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 . . .
And We Noted These Examples
If Sally likes Bill then Sally likes Bill.
Sally likes Bill’s mother, or not.
Sally likes Bill and Bill likes Jane, only if Bill likes Jane.
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 . . .
And We Noted These Examples
If Sally likes Bill then Sally likes Bill.
Sally likes Bill’s mother, or not.
Sally likes Bill and Bill likes Jane, only if Bill likes Jane.
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 . . .
Bill’s smart mother is a mother.
And We Noted These Examples
If Sally likes Bill then Sally likes Bill.
Sally likes Bill’s mother, or not.
Sally likes Bill and Bill likes Jane, only if Bill likes Jane.
…
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 . . .
Bill’s smart mother is a mother.
And We Noted These Examples
If Sally likes Bill then Sally likes Bill.
Sally likes Bill’s mother, or not.
Sally likes Bill and Bill likes Jane, only if Bill likes Jane.
…
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 . . .
Bill’s smart mother is a mother.
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
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 …
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 …
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 . . .
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 . . .
8x . . .
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 . . .
8x . . .
8✏(✏ > 0 ! 9�(� > 0 ^ 8x(d(x, a) < � ! d(f(x), b) < ✏)))
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 . . .
8x . . .
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 . . .
8x . . .
Every natural number is greater than or equal to zero.
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 . . .
8x . . .
Every natural number is greater than or equal to zero.8x(x � 0)
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 . . .
8x . . .
Every natural number is greater than or equal to zero.8x(x � 0)
There’s a positive integer greater than any positive integer.
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 . . .
8x . . .
Every natural number is greater than or equal to zero.8x(x � 0)
There’s a positive integer greater than any positive integer.9x8y(y < x)
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 . . .
8x . . .
Every natural number is greater than or equal to zero.8x(x � 0)
9x8y(y < x)
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 . . .
8x . . .
Every natural number is greater than or equal to zero.8x(x � 0)
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 . . .
8x . . .
Every natural number is greater than or equal to zero.8x(x � 0)
Every positive integer x is less-than-or-equal-to a positive integer y.
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 . . .
8x . . .
Every natural number is greater than or equal to zero.8x(x � 0)
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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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)).
<latexit sha1_base64="cUFmYnMZWq9+/1+rQoqL2aN6jL8=">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</latexit>
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
Tree for Run p1 ! p2<latexit sha1_base64="Bha6yGThqXjspNKfzw+UncF1azY=">AAACQHicbVDNSsNAGNzUv1r/WosnL4tF8FSSKuix6MVjBfsDbQibzaZdusmG3S9KKH0Wr/oavoVv4E28enLb5mBbBxaG+WZgdvxEcA22/WEVNja3tneKu6W9/YPDo3LluKNlqihrUymk6vlEM8Fj1gYOgvUSxUjkC9b1x3eze/eJKc1l/AhZwtyIDGMeckrASF65mngOHig+HAFRSj7jxGt45Zpdt+fA68TJSQ3laHkV62QQSJpGLAYqiNZ9x07AnRAFnAo2LQ1SzRJCx2TI+obGJGLanczbT/G5UQIcSmVeDHiu/k1MSKR1FvnGGREY6aWbhoioTAXT1cTM+m9ipigd6umaClIKvVIWwht3wuMkBRbTRdcwFRgknq2JA64YBZEZQqji5ruYjogiFMzmJTOkszrbOuk06s5l3Xm4qjVv80mL6BSdoQvkoGvURPeohdqIogy9oFf0Zr1bn9aX9b2wFqw8U0VLsH5+Aco2sAk=</latexit>
p1 true; p2 true<latexit sha1_base64="MMPfhJRyaxI+yvDPZaV/qDIMJUs=">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</latexit>
p1 false; p2 false<latexit sha1_base64="hhmvIh9/8CM3C4r0oZ/GNaJwkBU=">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</latexit>
p1 false; p2 true<latexit sha1_base64="mIqmUXithw8ZYQ1bjc+sJkBn1iU=">AAACY3icbVHLSiNBFK20OmqcGeNrJUIxQZhV6FZRwY3oxqWCUcEOTXXlthZWdTV1b4uh6T/xa9zqD/gB8x9TiVlo4oGCwzn3wL2n0kIrpDB8bwQzs3M/5hcWm0s/f/1ebq2sXqEtnYSutNq6m1QgaJVDlxRpuCkcCJNquE4fTof+9SM4VDa/pEEBPSPucpUpKchLSWu/SKKYxya1T1VM8EQoq0xohLo+inmR7Eya5ErvJa122AlH4NMkGpM2G+M8WWlsxH0rSwM5SS0Qb6OwoF4lHCmpoW7GJUIh5IO4g1tPc2EAe9XowJpve6XPM+v8y4mP1M+JShjEgUn9pBF0j188JCPcwPXrycRw9NvEUHGYYT2lkrUaJ5al7LBXqbwoCXL5sWtWak6WDwvnfeVAkh54IqRT/lwu74UTkvy3NH2R0WRt0+RqpxPtdqKLvfbxybjSBbbJ/rC/LGIH7JidsXPWZZI9sxf2yt4a/4KlYDVY/xgNGuPMGvuCYOs/Tb+78g==</latexit>
p3 true; p4 true<latexit sha1_base64="+HfvhwqHI/gXdTk99eGaOHSrLmQ=">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</latexit>
p3 true; p4 true<latexit sha1_base64="+HfvhwqHI/gXdTk99eGaOHSrLmQ=">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</latexit>
p3 true; p4 true<latexit sha1_base64="+HfvhwqHI/gXdTk99eGaOHSrLmQ=">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</latexit>
p3 ^ p4<latexit sha1_base64="gPEvciS6TeyWEcz03bxUojFwR/c=">AAACPXicbVDNSsNAGNzUv1r/ouLJy2IRPJXECnoUvXisYFVoSthsvrRLN9mw+0UJpW/iVV/D5/ABvIlXr25rD9o6sDDMNwOzE+VSGPS8N6eysLi0vFJdra2tb2xuuds7t0YVmkObK6n0fcQMSJFBGwVKuM81sDSScBcNLsf3uwfQRqjsBsscuinrZSIRnKGVQtfNw2ZAg0eIexDQPDwJ3brX8Cag88SfkjqZohVuO3tBrHiRQoZcMmM6vpdjd8g0Ci5hVAsKAznjA9aDjqUZS8F0h5PqI3polZgmStuXIZ2ovxNDlhpTppF1pgz75s/NYMp0qePRbGJs/TcxVrRJzGhORaWkmSmLyVl3KLK8QMj4T9ekkBQVHU9JY6GBoywtYVwL+13K+0wzjnbwmh3Sn51tntweN/xmw78+qZ9fTCetkn1yQI6IT07JObkiLdImnDyQJ/JMXpxX5935cD5/rBVnmtklf+B8fQOXTK5q</latexit>
p2 false<latexit sha1_base64="qqCu/uMlMN3AbCo7B6rj9KFfmIE=">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</latexit>
p2 false<latexit sha1_base64="qqCu/uMlMN3AbCo7B6rj9KFfmIE=">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</latexit>
p2 false<latexit sha1_base64="qqCu/uMlMN3AbCo7B6rj9KFfmIE=">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</latexit>
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=">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</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=">AAACY3icbVHLSiNBFK306KhxnImvlQjFBMFV6B5FhdmIblwqGBXs0FRXbmthVVdT97YYmv4Tv8at/oAf4H9YiVlo4oGCwzn3wL2n0kIrpDB8bQQ/ZmZ/zs0vNBd/Lf3+01peuUBbOgldabV1V6lA0CqHLinScFU4ECbVcJneHQ/9y3twqGx+ToMCekbc5CpTUpCXktZekezGPDapfahiggdCWWVCI9T1/5gXyd6kSa70XtJqh51wBD5NojFpszFOk+XGety3sjSQk9QC8ToKC+pVwpGSGupmXCIUQt6JG7j2NBcGsFeNDqz5llf6PLPOv5z4SP2cqIRBHJjUTxpBt/jFQzLCDVy/nkwMR79NDBWHGdZTKlmrcWJZyg56lcqLkiCXH7tmpeZk+bBw3lcOJOmBJ0I65c/l8lY4Icl/S9MXGU3WNk0u/nWinU50tts+PBpXOs822F+2zSK2zw7ZCTtlXSbZI3tiz+yl8RYsBivB2sdo0BhnVtkXBJvvWyi7+Q==</latexit>
p2 true<latexit sha1_base64="0VeFMQgaXI9uQSLSTJWi+dp6X/s=">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</latexit>
p2 true<latexit sha1_base64="0VeFMQgaXI9uQSLSTJWi+dp6X/s=">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</latexit>
p2 true<latexit sha1_base64="0VeFMQgaXI9uQSLSTJWi+dp6X/s=">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</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>
p4 true; p6 true<latexit sha1_base64="NedoU/0g7uIvVhvnrqrotOuceSA=">AAACYnicbVHLSgMxFE3HV62vVnGli2ARXJUZ3+Cm6MalglXBKUMmvaOhyWRI7ohlmC/xa9zqF7j3Q0xrF9p6IHA45x649yTOpLDo+58Vb2Z2bn6hulhbWl5ZXas31m+tzg2HDtdSm/uYWZAihQ4KlHCfGWAqlnAX9y+G/t0zGCt0eoODDLqKPaYiEZyhk6L6URYdhjRUsX4pQoQXtLxAk0NZnoU0i47/96J602/5I9BpEoxJk4xxFTUqm2FP81xBilwyax8CP8NuwQwKLqGshbmFjPE+e4QHR1OmwHaL0X0l3XVKjybauJciHam/EwVT1g5U7CYVwyf7x7OomBmYXjmZGI7+mxgqxia2nFJRa2knlsXktFuINMsRUv6za5JLipoO+6Y9YYCjHDjCuBHuXMqfmGEc3a/UXJHBZG3T5Ha/FRy0guvDZvt8XGmVbJEdskcCckLa5JJckQ7h5JW8kXfyUfnyal7D2/gZ9SrjzAb5A2/7G545u6Q=</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>
p5<latexit sha1_base64="zQBKeE+dacApeLKvft6OfbOe7FQ=">AAACLnicbVBLS8NAGNz4rPXVKp68LBbBU0l8oMeiF48V7QPaUDabTbt0kw27X4RQ8hO86t/w1wgexKs/w02bg20dWBjmm4HZ8WLBNdj2p7Wyura+sVnaKm/v7O7tV6oHbS0TRVmLSiFV1yOaCR6xFnAQrBsrRkJPsI43vsvvnWemNJfRE6Qxc0MyjHjAKQEjPcaDq0GlZtftKfAycQpSQwWag6p11PclTUIWARVE655jx+BOiAJOBcvK/USzmNAxGbKeoREJmXYn064ZPjWKjwOpzIsAT9W/iQkJtU5DzzhDAiM9d9MQEpUqP1tM5NZ/E7midKCzJRWkFHqhLAQ37oRHcQIsorOuQSIwSJxvh32uGAWRGkKo4ua7mI6IIhTMwmUzpLM42zJpn9edi7rzcFlr3BaTltAxOkFnyEHXqIHuURO1EEVD9IJe0Zv1bn1YX9b3zLpiFZlDNAfr5xc+H6lW</latexit>
p5 true<latexit sha1_base64="JvTnUEVxEj792Nvr/NVgyi0Yorc=">AAACRnicbVBNSyNBEK2J68fGj00UD8temg2CpzDjKnoM7mWPChsTcELo6dRoY/f00F0jCcPgr/Gqf8O/sH9ib7LX7cQc1mQfFDxevQdVL8mVdBSGv4LayofVtfWNj/XNre2dT43m7pUzhRXYFUYZ20+4QyUz7JIkhf3cIteJwl5y9326792jddJkP2mS40Dzm0ymUnDy0rDxOR+exCzWiRmXMeGYnCjJFlhVw0YrbIczsGUSzUkL5rgYNoP9eGREoTEjobhz11GY06DklqRQWNXjwmHOxR2/wWtPM67RDcrZDxU78MqIpcb6yYjN1H8TJdfOTXTinZrTrXu3c6S5ndhRtZiYWv+bmCrWpa5aUskY5RaOpfRsUMosLwgz8XZrWihGhk07ZSNpUZCaeMKFlf5dJm655YJ883VfZLRY2zK5OmpH39rR5XGrcz6vdAO+wFc4hAhOoQM/4AK6IOABHuEJnoOX4HfwGvx5s9aCeWYP3qEGfwHKoLKZ</latexit>
p5 true<latexit sha1_base64="JvTnUEVxEj792Nvr/NVgyi0Yorc=">AAACRnicbVBNSyNBEK2J68fGj00UD8temg2CpzDjKnoM7mWPChsTcELo6dRoY/f00F0jCcPgr/Gqf8O/sH9ib7LX7cQc1mQfFDxevQdVL8mVdBSGv4LayofVtfWNj/XNre2dT43m7pUzhRXYFUYZ20+4QyUz7JIkhf3cIteJwl5y9326792jddJkP2mS40Dzm0ymUnDy0rDxOR+exCzWiRmXMeGYnCjJFlhVw0YrbIczsGUSzUkL5rgYNoP9eGREoTEjobhz11GY06DklqRQWNXjwmHOxR2/wWtPM67RDcrZDxU78MqIpcb6yYjN1H8TJdfOTXTinZrTrXu3c6S5ndhRtZiYWv+bmCrWpa5aUskY5RaOpfRsUMosLwgz8XZrWihGhk07ZSNpUZCaeMKFlf5dJm655YJ883VfZLRY2zK5OmpH39rR5XGrcz6vdAO+wFc4hAhOoQM/4AK6IOABHuEJnoOX4HfwGvx5s9aCeWYP3qEGfwHKoLKZ</latexit>
p5 true<latexit sha1_base64="JvTnUEVxEj792Nvr/NVgyi0Yorc=">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</latexit>
p5 true<latexit sha1_base64="JvTnUEVxEj792Nvr/NVgyi0Yorc=">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</latexit>
p5 true<latexit sha1_base64="JvTnUEVxEj792Nvr/NVgyi0Yorc=">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</latexit>
p5 true<latexit sha1_base64="JvTnUEVxEj792Nvr/NVgyi0Yorc=">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</latexit>
p5 true<latexit sha1_base64="JvTnUEVxEj792Nvr/NVgyi0Yorc=">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</latexit>
p5 true<latexit sha1_base64="JvTnUEVxEj792Nvr/NVgyi0Yorc=">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</latexit>
p5 true<latexit sha1_base64="JvTnUEVxEj792Nvr/NVgyi0Yorc=">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</latexit>
...<latexit sha1_base64="dm9mWrB+txRWp3da3XJ47v4PeA4=">AAACMXicbVDNSsNAGNz4W+tfq3jyEiyCp5KooMeiF48V7A+0oWw2m3btbjbsfimEknfwqq/h0/QmXn0JN20OtnVgYZhvBmbHjznT4Dgza2Nza3tnt7RX3j84PDquVE/aWiaK0BaRXKqujzXlLKItYMBpN1YUC5/Tjj9+zO+dCVWayegF0ph6Ag8jFjKCwUjt/iSQoAeVmlN35rDXiVuQGirQHFSts34gSSJoBIRjrXuuE4M3xQoY4TQr9xNNY0zGeEh7hkZYUO1N53Uz+9IogR1KZV4E9lz9m5hioXUqfOMUGEZ66aZBYJWqIFtN5NZ/E7midKizNRWk5HqlLIT33pRFcQI0IouuYcJtkHY+nx0wRQnw1BBMFDPftckIK0zAjFw2Q7qrs62T9nXdvam7z7e1xkMxaQmdowt0hVx0hxroCTVRCxH0it7QO/qwPq2Z9WV9L6wbVpE5RUuwfn4BbPmq/A==</latexit>
...<latexit sha1_base64="dm9mWrB+txRWp3da3XJ47v4PeA4=">AAACMXicbVDNSsNAGNz4W+tfq3jyEiyCp5KooMeiF48V7A+0oWw2m3btbjbsfimEknfwqq/h0/QmXn0JN20OtnVgYZhvBmbHjznT4Dgza2Nza3tnt7RX3j84PDquVE/aWiaK0BaRXKqujzXlLKItYMBpN1YUC5/Tjj9+zO+dCVWayegF0ph6Ag8jFjKCwUjt/iSQoAeVmlN35rDXiVuQGirQHFSts34gSSJoBIRjrXuuE4M3xQoY4TQr9xNNY0zGeEh7hkZYUO1N53Uz+9IogR1KZV4E9lz9m5hioXUqfOMUGEZ66aZBYJWqIFtN5NZ/E7midKizNRWk5HqlLIT33pRFcQI0IouuYcJtkHY+nx0wRQnw1BBMFDPftckIK0zAjFw2Q7qrs62T9nXdvam7z7e1xkMxaQmdowt0hVx0hxroCTVRCxH0it7QO/qwPq2Z9WV9L6wbVpE5RUuwfn4BbPmq/A==</latexit>
...<latexit sha1_base64="dm9mWrB+txRWp3da3XJ47v4PeA4=">AAACMXicbVDNSsNAGNz4W+tfq3jyEiyCp5KooMeiF48V7A+0oWw2m3btbjbsfimEknfwqq/h0/QmXn0JN20OtnVgYZhvBmbHjznT4Dgza2Nza3tnt7RX3j84PDquVE/aWiaK0BaRXKqujzXlLKItYMBpN1YUC5/Tjj9+zO+dCVWayegF0ph6Ag8jFjKCwUjt/iSQoAeVmlN35rDXiVuQGirQHFSts34gSSJoBIRjrXuuE4M3xQoY4TQr9xNNY0zGeEh7hkZYUO1N53Uz+9IogR1KZV4E9lz9m5hioXUqfOMUGEZ66aZBYJWqIFtN5NZ/E7midKizNRWk5HqlLIT33pRFcQI0IouuYcJtkHY+nx0wRQnw1BBMFDPftckIK0zAjFw2Q7qrs62T9nXdvam7z7e1xkMxaQmdowt0hVx0hxroCTVRCxH0it7QO/qwPq2Z9WV9L6wbVpE5RUuwfn4BbPmq/A==</latexit>
...<latexit sha1_base64="dm9mWrB+txRWp3da3XJ47v4PeA4=">AAACMXicbVDNSsNAGNz4W+tfq3jyEiyCp5KooMeiF48V7A+0oWw2m3btbjbsfimEknfwqq/h0/QmXn0JN20OtnVgYZhvBmbHjznT4Dgza2Nza3tnt7RX3j84PDquVE/aWiaK0BaRXKqujzXlLKItYMBpN1YUC5/Tjj9+zO+dCVWayegF0ph6Ag8jFjKCwUjt/iSQoAeVmlN35rDXiVuQGirQHFSts34gSSJoBIRjrXuuE4M3xQoY4TQr9xNNY0zGeEh7hkZYUO1N53Uz+9IogR1KZV4E9lz9m5hioXUqfOMUGEZ66aZBYJWqIFtN5NZ/E7midKizNRWk5HqlLIT33pRFcQI0IouuYcJtkHY+nx0wRQnw1BBMFDPftckIK0zAjFw2Q7qrs62T9nXdvam7z7e1xkMxaQmdowt0hVx0hxroCTVRCxH0it7QO/qwPq2Z9WV9L6wbVpE5RUuwfn4BbPmq/A==</latexit>
...<latexit sha1_base64="dm9mWrB+txRWp3da3XJ47v4PeA4=">AAACMXicbVDNSsNAGNz4W+tfq3jyEiyCp5KooMeiF48V7A+0oWw2m3btbjbsfimEknfwqq/h0/QmXn0JN20OtnVgYZhvBmbHjznT4Dgza2Nza3tnt7RX3j84PDquVE/aWiaK0BaRXKqujzXlLKItYMBpN1YUC5/Tjj9+zO+dCVWayegF0ph6Ag8jFjKCwUjt/iSQoAeVmlN35rDXiVuQGirQHFSts34gSSJoBIRjrXuuE4M3xQoY4TQr9xNNY0zGeEh7hkZYUO1N53Uz+9IogR1KZV4E9lz9m5hioXUqfOMUGEZ66aZBYJWqIFtN5NZ/E7midKizNRWk5HqlLIT33pRFcQI0IouuYcJtkHY+nx0wRQnw1BBMFDPftckIK0zAjFw2Q7qrs62T9nXdvam7z7e1xkMxaQmdowt0hVx0hxroCTVRCxH0it7QO/qwPq2Z9WV9L6wbVpE5RUuwfn4BbPmq/A==</latexit>
...<latexit sha1_base64="dm9mWrB+txRWp3da3XJ47v4PeA4=">AAACMXicbVDNSsNAGNz4W+tfq3jyEiyCp5KooMeiF48V7A+0oWw2m3btbjbsfimEknfwqq/h0/QmXn0JN20OtnVgYZhvBmbHjznT4Dgza2Nza3tnt7RX3j84PDquVE/aWiaK0BaRXKqujzXlLKItYMBpN1YUC5/Tjj9+zO+dCVWayegF0ph6Ag8jFjKCwUjt/iSQoAeVmlN35rDXiVuQGirQHFSts34gSSJoBIRjrXuuE4M3xQoY4TQr9xNNY0zGeEh7hkZYUO1N53Uz+9IogR1KZV4E9lz9m5hioXUqfOMUGEZ66aZBYJWqIFtN5NZ/E7midKizNRWk5HqlLIT33pRFcQI0IouuYcJtkHY+nx0wRQnw1BBMFDPftckIK0zAjFw2Q7qrs62T9nXdvam7z7e1xkMxaQmdowt0hVx0hxroCTVRCxH0it7QO/qwPq2Z9WV9L6wbVpE5RUuwfn4BbPmq/A==</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?
Exercise 2: Is there an algorithm for traveling this way?
No. This strategy for travel is beyond the reach of standard computation.
Exercise 2: Is there an algorithm for traveling this way?
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
For Further Reading
For Further Reading
For Further Reading
For Further Reading
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
Needs Understanding of Ordinal Numbers …
191 = 2222
0
+ 220
+ 20 < !!!!0
+ !!0
+ !0
Needs Understanding of Ordinal Numbers …
191 = 2222
0
+ 220
+ 20 < !!!!0
+ !!0
+ !0
192 = 3333
0
+ 330
+ 30 � 1 < !!!!0
+ !!0
+ !0 � 1
Needs Understanding of Ordinal Numbers …
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
Needs Understanding of Ordinal Numbers …
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
Needs Understanding of Ordinal Numbers …
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
Needs Understanding of Ordinal Numbers …
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
...
Needs Understanding of Ordinal Numbers …
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
...
Needs Understanding of Ordinal Numbers …
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
Needs Understanding of Ordinal Numbers …
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”)
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.
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.
If this is right, and computing machines can’t use irreducibly infinitary techniques, they’re in trouble — or: there won’t be a Singularity.