The Secret Life of The Second Incompleteness Inconsistency - PowerPoint PPT Presentation

Beginning The Secret Life of The Second Incompleteness Inconsistency Statements Theorem Philosophical Worries From Bad Philosophy to Cool Albert Visser Mathematics Devices and Teleports Department of Philosophy, Faculty of Humanities, Utrecht University ALCOP 2013 April 20, 2013, Utrecht 1

Overview Beginning Beginning The Second Incompleteness Theorem Philosophical The Second Incompleteness Theorem Worries From Bad Philosophy to Cool Mathematics Philosophical Worries Devices and Teleports From Bad Philosophy to Cool Mathematics Devices and Teleports 2

Overview Beginning Beginning The Second Incompleteness Theorem Philosophical The Second Incompleteness Theorem Worries From Bad Philosophy to Cool Mathematics Philosophical Worries Devices and Teleports From Bad Philosophy to Cool Mathematics Devices and Teleports 2

Overview Beginning Beginning The Second Incompleteness Theorem Philosophical The Second Incompleteness Theorem Worries From Bad Philosophy to Cool Mathematics Philosophical Worries Devices and Teleports From Bad Philosophy to Cool Mathematics Devices and Teleports 2

Overview Beginning Beginning The Second Incompleteness Theorem Philosophical The Second Incompleteness Theorem Worries From Bad Philosophy to Cool Mathematics Philosophical Worries Devices and Teleports From Bad Philosophy to Cool Mathematics Devices and Teleports 2

Overview Beginning Beginning The Second Incompleteness Theorem Philosophical The Second Incompleteness Theorem Worries From Bad Philosophy to Cool Mathematics Philosophical Worries Devices and Teleports From Bad Philosophy to Cool Mathematics Devices and Teleports 2

Overview Beginning Beginning The Second Incompleteness Theorem Philosophical The Second Incompleteness Theorem Worries From Bad Philosophy to Cool Mathematics Philosophical Worries Devices and Teleports From Bad Philosophy to Cool Mathematics Devices and Teleports 3

Limits and Scope Beginning The Second Incompleteness Theorem Philosophical Worries From Bad The Limits and Scope of Mathematical Knowledge Philosophy to Cool Mathematics Bristol, March 30+31 Devices and Teleports 4

Overview Beginning Beginning The Second Incompleteness Theorem Philosophical The Second Incompleteness Theorem Worries From Bad Philosophy to Cool Mathematics Philosophical Worries Devices and Teleports From Bad Philosophy to Cool Mathematics Devices and Teleports 5

Interpretability Formulations of G2: 1 We write U ✄ V for: U interprets V , and V ✁ U for: V is interrpetable in U . The theory PA − is the theory of discretely Beginning The Second ordered commutative semirings with a least element. Incompleteness Theorem Philosophical For any consistent CE theory U , we have: Worries ◮ U � ✄ ( PA − + con ( U )) . From Bad Philosophy to Cool Mathematics This version rests on ideas of Solovay, Friedman, Pudlák. Note Devices and Teleports that we really should have written con ( α U ) , where α U is (at most) a Σ 1 -representation of the axioms of U . Instead of PA − we could as well have taken Q or S 1 2 or I ∆ 0 + Ω 1 . Moreover, instead of an arithmetical theory we could have employed a string theory like Grzegorczyk’s theory TC or a theory of (possibly finite) sets like Adjunctive Set Theory AS. 6

Interpretability Formulations of G2: 1 We write U ✄ V for: U interprets V , and V ✁ U for: V is interrpetable in U . The theory PA − is the theory of discretely Beginning The Second ordered commutative semirings with a least element. Incompleteness Theorem Philosophical For any consistent CE theory U , we have: Worries ◮ U � ✄ ( PA − + con ( U )) . From Bad Philosophy to Cool Mathematics This version rests on ideas of Solovay, Friedman, Pudlák. Note Devices and Teleports that we really should have written con ( α U ) , where α U is (at most) a Σ 1 -representation of the axioms of U . Instead of PA − we could as well have taken Q or S 1 2 or I ∆ 0 + Ω 1 . Moreover, instead of an arithmetical theory we could have employed a string theory like Grzegorczyk’s theory TC or a theory of (possibly finite) sets like Adjunctive Set Theory AS. 6

Interpretability Formulations of G2: 1 We write U ✄ V for: U interprets V , and V ✁ U for: V is interrpetable in U . The theory PA − is the theory of discretely Beginning The Second ordered commutative semirings with a least element. Incompleteness Theorem Philosophical For any consistent CE theory U , we have: Worries ◮ U � ✄ ( PA − + con ( U )) . From Bad Philosophy to Cool Mathematics This version rests on ideas of Solovay, Friedman, Pudlák. Note Devices and Teleports that we really should have written con ( α U ) , where α U is (at most) a Σ 1 -representation of the axioms of U . Instead of PA − we could as well have taken Q or S 1 2 or I ∆ 0 + Ω 1 . Moreover, instead of an arithmetical theory we could have employed a string theory like Grzegorczyk’s theory TC or a theory of (possibly finite) sets like Adjunctive Set Theory AS. 6

Interpretability Formulations of G2: 1 We write U ✄ V for: U interprets V , and V ✁ U for: V is interrpetable in U . The theory PA − is the theory of discretely Beginning The Second ordered commutative semirings with a least element. Incompleteness Theorem Philosophical For any consistent CE theory U , we have: Worries ◮ U � ✄ ( PA − + con ( U )) . From Bad Philosophy to Cool Mathematics This version rests on ideas of Solovay, Friedman, Pudlák. Note Devices and Teleports that we really should have written con ( α U ) , where α U is (at most) a Σ 1 -representation of the axioms of U . Instead of PA − we could as well have taken Q or S 1 2 or I ∆ 0 + Ω 1 . Moreover, instead of an arithmetical theory we could have employed a string theory like Grzegorczyk’s theory TC or a theory of (possibly finite) sets like Adjunctive Set Theory AS. 6

Interpretability Formulations of G2: 1 We write U ✄ V for: U interprets V , and V ✁ U for: V is interrpetable in U . The theory PA − is the theory of discretely Beginning The Second ordered commutative semirings with a least element. Incompleteness Theorem Philosophical For any consistent CE theory U , we have: Worries ◮ U � ✄ ( PA − + con ( U )) . From Bad Philosophy to Cool Mathematics This version rests on ideas of Solovay, Friedman, Pudlák. Note Devices and Teleports that we really should have written con ( α U ) , where α U is (at most) a Σ 1 -representation of the axioms of U . Instead of PA − we could as well have taken Q or S 1 2 or I ∆ 0 + Ω 1 . Moreover, instead of an arithmetical theory we could have employed a string theory like Grzegorczyk’s theory TC or a theory of (possibly finite) sets like Adjunctive Set Theory AS. 6

Interpretability Formulations of G2: 2 Combining the above version with a strong variant of the Beginning Gödel-Hilbert-Bernays-Wang-Henkin-Feferman Theorem, we get The Second Incompleteness an even more appealing form. For CE U , we have, writing ⊤ ⊤ for Theorem Philosophical the inconsistent theory: Worries � = ( PA − + con ( U )) ✁ ◮ If U ✁ ⊤ , then U ✁ ⊤ . � = ⊤ � = ⊤ From Bad Philosophy to Cool Mathematics Given a theory U of signature Σ and an arbitrary second signature Devices and Teleports Θ . An inference A ∼ U B is ( Θ -)admissible if, for all translations τ : Θ → Σ , we have U ⊢ A τ ⇒ U ⊢ B τ . G2 yields a non-trivial admissible rule, for each CE theory U : ( PA − + con ( U )) ∼ U ⊥ . 7

Interpretability Formulations of G2: 2 Combining the above version with a strong variant of the Beginning Gödel-Hilbert-Bernays-Wang-Henkin-Feferman Theorem, we get The Second Incompleteness an even more appealing form. For CE U , we have, writing ⊤ ⊤ for Theorem Philosophical the inconsistent theory: Worries � = ( PA − + con ( U )) ✁ ◮ If U ✁ ⊤ , then U ✁ ⊤ . � = ⊤ � = ⊤ From Bad Philosophy to Cool Mathematics Given a theory U of signature Σ and an arbitrary second signature Devices and Teleports Θ . An inference A ∼ U B is ( Θ -)admissible if, for all translations τ : Θ → Σ , we have U ⊢ A τ ⇒ U ⊢ B τ . G2 yields a non-trivial admissible rule, for each CE theory U : ( PA − + con ( U )) ∼ U ⊥ . 7

Interpretability Formulations of G2: 2 Combining the above version with a strong variant of the Beginning Gödel-Hilbert-Bernays-Wang-Henkin-Feferman Theorem, we get The Second Incompleteness an even more appealing form. For CE U , we have, writing ⊤ ⊤ for Theorem Philosophical the inconsistent theory: Worries � = ( PA − + con ( U )) ✁ ◮ If U ✁ ⊤ , then U ✁ ⊤ . � = ⊤ � = ⊤ From Bad Philosophy to Cool Mathematics Given a theory U of signature Σ and an arbitrary second signature Devices and Teleports Θ . An inference A ∼ U B is ( Θ -)admissible if, for all translations τ : Θ → Σ , we have U ⊢ A τ ⇒ U ⊢ B τ . G2 yields a non-trivial admissible rule, for each CE theory U : ( PA − + con ( U )) ∼ U ⊥ . 7