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

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

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The Secret Life of Inconsistency Statements

Albert Visser

Department of Philosophy, Faculty of Humanities, Utrecht University

ALCOP 2013 April 20, 2013, Utrecht

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

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Overview

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

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

2

Overview

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

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

2

Overview

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

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

2

Overview

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

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

2

Overview

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

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

3

Overview

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

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

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Limits and Scope

The Limits and Scope of Mathematical Knowledge Bristol, March 30+31

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

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Overview

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

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

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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

• rdered commutative semirings with a least element.

For any consistent CE theory U, we have:

◮ U ✄ (PA− + con(U)).

This version rests on ideas of Solovay, Friedman, Pudlák. Note that we really should have written con(αU), where αU is (at most) a Σ1-representation of the axioms of U. Instead of PA− we could as well have taken Q or S1

2 or I∆0 + Ω1.

Moreover, instead of an arithmetical theory we could have employed a string theory like Grzegorczyk’s theory TC or a theory

• f (possibly finite) sets like Adjunctive Set Theory AS.

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

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

• rdered commutative semirings with a least element.

For any consistent CE theory U, we have:

◮ U ✄ (PA− + con(U)).

This version rests on ideas of Solovay, Friedman, Pudlák. Note that we really should have written con(αU), where αU is (at most) a Σ1-representation of the axioms of U. Instead of PA− we could as well have taken Q or S1

2 or I∆0 + Ω1.

Moreover, instead of an arithmetical theory we could have employed a string theory like Grzegorczyk’s theory TC or a theory

• f (possibly finite) sets like Adjunctive Set Theory AS.

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

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

• rdered commutative semirings with a least element.

For any consistent CE theory U, we have:

◮ U ✄ (PA− + con(U)).

This version rests on ideas of Solovay, Friedman, Pudlák. Note that we really should have written con(αU), where αU is (at most) a Σ1-representation of the axioms of U. Instead of PA− we could as well have taken Q or S1

2 or I∆0 + Ω1.

Moreover, instead of an arithmetical theory we could have employed a string theory like Grzegorczyk’s theory TC or a theory

• f (possibly finite) sets like Adjunctive Set Theory AS.

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

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

• rdered commutative semirings with a least element.

For any consistent CE theory U, we have:

◮ U ✄ (PA− + con(U)).

This version rests on ideas of Solovay, Friedman, Pudlák. Note that we really should have written con(αU), where αU is (at most) a Σ1-representation of the axioms of U. Instead of PA− we could as well have taken Q or S1

2 or I∆0 + Ω1.

Moreover, instead of an arithmetical theory we could have employed a string theory like Grzegorczyk’s theory TC or a theory

• f (possibly finite) sets like Adjunctive Set Theory AS.

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

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

• rdered commutative semirings with a least element.

For any consistent CE theory U, we have:

◮ U ✄ (PA− + con(U)).

This version rests on ideas of Solovay, Friedman, Pudlák. Note that we really should have written con(αU), where αU is (at most) a Σ1-representation of the axioms of U. Instead of PA− we could as well have taken Q or S1

2 or I∆0 + Ω1.

Moreover, instead of an arithmetical theory we could have employed a string theory like Grzegorczyk’s theory TC or a theory

• f (possibly finite) sets like Adjunctive Set Theory AS.

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

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Interpretability Formulations of G2: 2

Combining the above version with a strong variant of the Gödel-Hilbert-Bernays-Wang-Henkin-Feferman Theorem, we get an even more appealing form. For CE U, we have, writing ⊤ ⊤ for the inconsistent theory:

◮ If U ✁ = ⊤

⊤, then U ✁

= (PA− + con(U)) ✁ = ⊤

⊤. Given a theory U of signature Σ and an arbitrary second signature Θ. An inference A

∼ U B is (Θ-)admissible if, for all translations

τ : Θ → Σ, we have U ⊢ Aτ ⇒ U ⊢ Bτ. G2 yields a non-trivial admissible rule, for each CE theory U: (PA− + con(U))

∼ U ⊥.

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

7

Interpretability Formulations of G2: 2

Combining the above version with a strong variant of the Gödel-Hilbert-Bernays-Wang-Henkin-Feferman Theorem, we get an even more appealing form. For CE U, we have, writing ⊤ ⊤ for the inconsistent theory:

◮ If U ✁ = ⊤

⊤, then U ✁

= (PA− + con(U)) ✁ = ⊤

⊤. Given a theory U of signature Σ and an arbitrary second signature Θ. An inference A

∼ U B is (Θ-)admissible if, for all translations

τ : Θ → Σ, we have U ⊢ Aτ ⇒ U ⊢ Bτ. G2 yields a non-trivial admissible rule, for each CE theory U: (PA− + con(U))

∼ U ⊥.

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

7

Interpretability Formulations of G2: 2

Combining the above version with a strong variant of the Gödel-Hilbert-Bernays-Wang-Henkin-Feferman Theorem, we get an even more appealing form. For CE U, we have, writing ⊤ ⊤ for the inconsistent theory:

◮ If U ✁ = ⊤

⊤, then U ✁

= (PA− + con(U)) ✁ = ⊤

⊤. Given a theory U of signature Σ and an arbitrary second signature Θ. An inference A

∼ U B is (Θ-)admissible if, for all translations

τ : Θ → Σ, we have U ⊢ Aτ ⇒ U ⊢ Bτ. G2 yields a non-trivial admissible rule, for each CE theory U: (PA− + con(U))

∼ U ⊥.

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

7

Interpretability Formulations of G2: 2

Combining the above version with a strong variant of the Gödel-Hilbert-Bernays-Wang-Henkin-Feferman Theorem, we get an even more appealing form. For CE U, we have, writing ⊤ ⊤ for the inconsistent theory:

◮ If U ✁ = ⊤

⊤, then U ✁

= (PA− + con(U)) ✁ = ⊤

⊤. Given a theory U of signature Σ and an arbitrary second signature Θ. An inference A

∼ U B is (Θ-)admissible if, for all translations

τ : Θ → Σ, we have U ⊢ Aτ ⇒ U ⊢ Bτ. G2 yields a non-trivial admissible rule, for each CE theory U: (PA− + con(U))

∼ U ⊥.

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

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Feferman’s Theorem

Here is my general formulation of the Feferman Theorem: N : U ✄ S1

2

⇒ U ✄ (U + inconN(U)). Proof: Clearly (i) id : (U + inconN(U)) ✄ (U + inconN(U)). Also (ii): (U + conN(U)) ⊢ (U + conN(U + inconN(U))) ✄ (U + inconN(U)). The last step uses the GBWHF theorem. Combining (i) and (ii) using a disjunctive interpretation we find: U ✄ (U + inconN(U)). ✷ Modal notations: ⊤ ✄U ✷N

U⊥ or ⊤ ✄U,N ✷U⊥.

If U and N are fixed in the background: ⊤ ✄ ✷⊥.

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

8

Feferman’s Theorem

Here is my general formulation of the Feferman Theorem: N : U ✄ S1

2

⇒ U ✄ (U + inconN(U)). Proof: Clearly (i) id : (U + inconN(U)) ✄ (U + inconN(U)). Also (ii): (U + conN(U)) ⊢ (U + conN(U + inconN(U))) ✄ (U + inconN(U)). The last step uses the GBWHF theorem. Combining (i) and (ii) using a disjunctive interpretation we find: U ✄ (U + inconN(U)). ✷ Modal notations: ⊤ ✄U ✷N

U⊥ or ⊤ ✄U,N ✷U⊥.

If U and N are fixed in the background: ⊤ ✄ ✷⊥.

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

8

Feferman’s Theorem

Here is my general formulation of the Feferman Theorem: N : U ✄ S1

2

⇒ U ✄ (U + inconN(U)). Proof: Clearly (i) id : (U + inconN(U)) ✄ (U + inconN(U)). Also (ii): (U + conN(U)) ⊢ (U + conN(U + inconN(U))) ✄ (U + inconN(U)). The last step uses the GBWHF theorem. Combining (i) and (ii) using a disjunctive interpretation we find: U ✄ (U + inconN(U)). ✷ Modal notations: ⊤ ✄U ✷N

U⊥ or ⊤ ✄U,N ✷U⊥.

If U and N are fixed in the background: ⊤ ✄ ✷⊥.

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

8

Feferman’s Theorem

Here is my general formulation of the Feferman Theorem: N : U ✄ S1

2

⇒ U ✄ (U + inconN(U)). Proof: Clearly (i) id : (U + inconN(U)) ✄ (U + inconN(U)). Also (ii): (U + conN(U)) ⊢ (U + conN(U + inconN(U))) ✄ (U + inconN(U)). The last step uses the GBWHF theorem. Combining (i) and (ii) using a disjunctive interpretation we find: U ✄ (U + inconN(U)). ✷ Modal notations: ⊤ ✄U ✷N

U⊥ or ⊤ ✄U,N ✷U⊥.

If U and N are fixed in the background: ⊤ ✄ ✷⊥.

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

8

Feferman’s Theorem

Here is my general formulation of the Feferman Theorem: N : U ✄ S1

2

⇒ U ✄ (U + inconN(U)). Proof: Clearly (i) id : (U + inconN(U)) ✄ (U + inconN(U)). Also (ii): (U + conN(U)) ⊢ (U + conN(U + inconN(U))) ✄ (U + inconN(U)). The last step uses the GBWHF theorem. Combining (i) and (ii) using a disjunctive interpretation we find: U ✄ (U + inconN(U)). ✷ Modal notations: ⊤ ✄U ✷N

U⊥ or ⊤ ✄U,N ✷U⊥.

If U and N are fixed in the background: ⊤ ✄ ✷⊥.

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

8

Feferman’s Theorem

Here is my general formulation of the Feferman Theorem: N : U ✄ S1

2

⇒ U ✄ (U + inconN(U)). Proof: Clearly (i) id : (U + inconN(U)) ✄ (U + inconN(U)). Also (ii): (U + conN(U)) ⊢ (U + conN(U + inconN(U))) ✄ (U + inconN(U)). The last step uses the GBWHF theorem. Combining (i) and (ii) using a disjunctive interpretation we find: U ✄ (U + inconN(U)). ✷ Modal notations: ⊤ ✄U ✷N

U⊥ or ⊤ ✄U,N ✷U⊥.

If U and N are fixed in the background: ⊤ ✄ ✷⊥.

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

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Overview

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

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

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A Question

What does it mean that a sentence A in the language of T is a consistency statement for T? The usual statements like con(T) cannot have their usual meanings if T is unsound and cannot have standard models. And what if T is a very weak theory like Q? In this case it cannot verify any properties of the usual arithmetization of the proof-predicate. Perhaps T should verify various properties of A or properties of the ingredients that A is constructed from? Specifically one could demand that the provability predicate has certain desirable

• properties. This reflects the idea that the meaning of the predicate

is determined by the role it plays in the reasoning in the theory.

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

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A Question

What does it mean that a sentence A in the language of T is a consistency statement for T? The usual statements like con(T) cannot have their usual meanings if T is unsound and cannot have standard models. And what if T is a very weak theory like Q? In this case it cannot verify any properties of the usual arithmetization of the proof-predicate. Perhaps T should verify various properties of A or properties of the ingredients that A is constructed from? Specifically one could demand that the provability predicate has certain desirable

• properties. This reflects the idea that the meaning of the predicate

is determined by the role it plays in the reasoning in the theory.

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

10

A Question

What does it mean that a sentence A in the language of T is a consistency statement for T? The usual statements like con(T) cannot have their usual meanings if T is unsound and cannot have standard models. And what if T is a very weak theory like Q? In this case it cannot verify any properties of the usual arithmetization of the proof-predicate. Perhaps T should verify various properties of A or properties of the ingredients that A is constructed from? Specifically one could demand that the provability predicate has certain desirable

• properties. This reflects the idea that the meaning of the predicate

is determined by the role it plays in the reasoning in the theory.

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

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L-predicates 1

We consider one possible explication of the above answer. We fix a base theory T. A consistency statement is the consistency statement for an L-predicate △. This means: A has the form ¬ △T⊥, where △ := △T satisfies the Löb conditions w.r.t. T.

• L1. ⊢ A

⇒ ⊢ △A

• L2. ⊢ (△A ∧ △(A → B)) → △B
• L3. ⊢ △A → △△A

We note that x = x is an L-predicate. This makes ⊥ = ⊥ a consistency statement.

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

11

L-predicates 1

We consider one possible explication of the above answer. We fix a base theory T. A consistency statement is the consistency statement for an L-predicate △. This means: A has the form ¬ △T⊥, where △ := △T satisfies the Löb conditions w.r.t. T.

• L1. ⊢ A

⇒ ⊢ △A

• L2. ⊢ (△A ∧ △(A → B)) → △B
• L3. ⊢ △A → △△A

We note that x = x is an L-predicate. This makes ⊥ = ⊥ a consistency statement.

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

11

L-predicates 1

We consider one possible explication of the above answer. We fix a base theory T. A consistency statement is the consistency statement for an L-predicate △. This means: A has the form ¬ △T⊥, where △ := △T satisfies the Löb conditions w.r.t. T.

• L1. ⊢ A

⇒ ⊢ △A

• L2. ⊢ (△A ∧ △(A → B)) → △B
• L3. ⊢ △A → △△A

We note that x = x is an L-predicate. This makes ⊥ = ⊥ a consistency statement.

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

11

L-predicates 1

We consider one possible explication of the above answer. We fix a base theory T. A consistency statement is the consistency statement for an L-predicate △. This means: A has the form ¬ △T⊥, where △ := △T satisfies the Löb conditions w.r.t. T.

• L1. ⊢ A

⇒ ⊢ △A

• L2. ⊢ (△A ∧ △(A → B)) → △B
• L3. ⊢ △A → △△A

We note that x = x is an L-predicate. This makes ⊥ = ⊥ a consistency statement.

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

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L-predicates 2

Can we accept ⊥ = ⊥ as a possible consistency statement? To be a consistency statement is in this conception not an absolute concept but a role w.r.t. a choice of an L-predicate. This strategy has a Hilbertian feel to it. We formulate G2 as: for all consistent theories T that interpret, say, R via K and all L-predicates △ for T, K, we have T ¬ △⊥, where the gödel numbering is a standard one, used relative to K. It seems to me that this radical answer cannot satisfy the foundationally motivated questioner who wants to lay down conditions for what it is to express consistency. We precisely want to link the consistency statements with our ordinary understanding

• f consistency and this is completely lost if consistency is

⊥ = ⊥. Further moves: putting more restrictions on the predicate like the Kreisel condition: ⊢ △A ⇒ ⊢ A.

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

12

L-predicates 2

Can we accept ⊥ = ⊥ as a possible consistency statement? To be a consistency statement is in this conception not an absolute concept but a role w.r.t. a choice of an L-predicate. This strategy has a Hilbertian feel to it. We formulate G2 as: for all consistent theories T that interpret, say, R via K and all L-predicates △ for T, K, we have T ¬ △⊥, where the gödel numbering is a standard one, used relative to K. It seems to me that this radical answer cannot satisfy the foundationally motivated questioner who wants to lay down conditions for what it is to express consistency. We precisely want to link the consistency statements with our ordinary understanding

• f consistency and this is completely lost if consistency is

⊥ = ⊥. Further moves: putting more restrictions on the predicate like the Kreisel condition: ⊢ △A ⇒ ⊢ A.

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

12

L-predicates 2

Can we accept ⊥ = ⊥ as a possible consistency statement? To be a consistency statement is in this conception not an absolute concept but a role w.r.t. a choice of an L-predicate. This strategy has a Hilbertian feel to it. We formulate G2 as: for all consistent theories T that interpret, say, R via K and all L-predicates △ for T, K, we have T ¬ △⊥, where the gödel numbering is a standard one, used relative to K. It seems to me that this radical answer cannot satisfy the foundationally motivated questioner who wants to lay down conditions for what it is to express consistency. We precisely want to link the consistency statements with our ordinary understanding

• f consistency and this is completely lost if consistency is

⊥ = ⊥. Further moves: putting more restrictions on the predicate like the Kreisel condition: ⊢ △A ⇒ ⊢ A.

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

12

L-predicates 2

Can we accept ⊥ = ⊥ as a possible consistency statement? To be a consistency statement is in this conception not an absolute concept but a role w.r.t. a choice of an L-predicate. This strategy has a Hilbertian feel to it. We formulate G2 as: for all consistent theories T that interpret, say, R via K and all L-predicates △ for T, K, we have T ¬ △⊥, where the gödel numbering is a standard one, used relative to K. It seems to me that this radical answer cannot satisfy the foundationally motivated questioner who wants to lay down conditions for what it is to express consistency. We precisely want to link the consistency statements with our ordinary understanding

• f consistency and this is completely lost if consistency is

⊥ = ⊥. Further moves: putting more restrictions on the predicate like the Kreisel condition: ⊢ △A ⇒ ⊢ A.

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

12

L-predicates 2

Can we accept ⊥ = ⊥ as a possible consistency statement? To be a consistency statement is in this conception not an absolute concept but a role w.r.t. a choice of an L-predicate. This strategy has a Hilbertian feel to it. We formulate G2 as: for all consistent theories T that interpret, say, R via K and all L-predicates △ for T, K, we have T ¬ △⊥, where the gödel numbering is a standard one, used relative to K. It seems to me that this radical answer cannot satisfy the foundationally motivated questioner who wants to lay down conditions for what it is to express consistency. We precisely want to link the consistency statements with our ordinary understanding

• f consistency and this is completely lost if consistency is

⊥ = ⊥. Further moves: putting more restrictions on the predicate like the Kreisel condition: ⊢ △A ⇒ ⊢ A.

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

13

Deconstruction

We remind the reader of our earlier formulation of G2: For any consistent CE theory U, we have:

◮ U ✄ (PA− + con(U)).

My preference is to deconstruct the question. G2 is best viewed as a claim about a relation between theories. It is a claim about the interpretability relation between an arbitrary theory T —viewed as merely formal— and the meaningful theory PA− + con(T); even better: a corresponding meaningful syntactic theory. This theory is considered to have the standard semantics. At no point properties

• f T are needed to give meaning to con(T).

This does not mean that there is no problem of what it is to express consistency, e.g. when we compare ordinary consistency with Feferman consistency. However, this question is asked against the background of the standard semantics.

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

13

Deconstruction

We remind the reader of our earlier formulation of G2: For any consistent CE theory U, we have:

◮ U ✄ (PA− + con(U)).

My preference is to deconstruct the question. G2 is best viewed as a claim about a relation between theories. It is a claim about the interpretability relation between an arbitrary theory T —viewed as merely formal— and the meaningful theory PA− + con(T); even better: a corresponding meaningful syntactic theory. This theory is considered to have the standard semantics. At no point properties

• f T are needed to give meaning to con(T).

This does not mean that there is no problem of what it is to express consistency, e.g. when we compare ordinary consistency with Feferman consistency. However, this question is asked against the background of the standard semantics.

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

13

Deconstruction

We remind the reader of our earlier formulation of G2: For any consistent CE theory U, we have:

◮ U ✄ (PA− + con(U)).

My preference is to deconstruct the question. G2 is best viewed as a claim about a relation between theories. It is a claim about the interpretability relation between an arbitrary theory T —viewed as merely formal— and the meaningful theory PA− + con(T); even better: a corresponding meaningful syntactic theory. This theory is considered to have the standard semantics. At no point properties

• f T are needed to give meaning to con(T).

This does not mean that there is no problem of what it is to express consistency, e.g. when we compare ordinary consistency with Feferman consistency. However, this question is asked against the background of the standard semantics.

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

13

Deconstruction

We remind the reader of our earlier formulation of G2: For any consistent CE theory U, we have:

◮ U ✄ (PA− + con(U)).

My preference is to deconstruct the question. G2 is best viewed as a claim about a relation between theories. It is a claim about the interpretability relation between an arbitrary theory T —viewed as merely formal— and the meaningful theory PA− + con(T); even better: a corresponding meaningful syntactic theory. This theory is considered to have the standard semantics. At no point properties

• f T are needed to give meaning to con(T).

This does not mean that there is no problem of what it is to express consistency, e.g. when we compare ordinary consistency with Feferman consistency. However, this question is asked against the background of the standard semantics.

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

14

Overview

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

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

15

Alternative Motivation

The fact that the relativization to HBL-predicates approach to meaning does not hold water, does not mean that such versions of G2 are not mathematically interesting. We will connect the approach to:

◮ Developing the analogue of model theory: a HBL-predicate is

like a model of (an extension of) Löb’s Logic.

◮ Finding a converse of Feferman’s Theorem. ◮ Finding a new semantics for interpretability logic.

Note that there is also the perspective of extending theories with new modal predicates as is done in Epistemic Arithmetic.

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

15

Alternative Motivation

The fact that the relativization to HBL-predicates approach to meaning does not hold water, does not mean that such versions of G2 are not mathematically interesting. We will connect the approach to:

◮ Developing the analogue of model theory: a HBL-predicate is

like a model of (an extension of) Löb’s Logic.

◮ Finding a converse of Feferman’s Theorem. ◮ Finding a new semantics for interpretability logic.

Note that there is also the perspective of extending theories with new modal predicates as is done in Epistemic Arithmetic.

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

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Alternative Motivation

The fact that the relativization to HBL-predicates approach to meaning does not hold water, does not mean that such versions of G2 are not mathematically interesting. We will connect the approach to:

◮ Developing the analogue of model theory: a HBL-predicate is

like a model of (an extension of) Löb’s Logic.

◮ Finding a converse of Feferman’s Theorem. ◮ Finding a new semantics for interpretability logic.

Note that there is also the perspective of extending theories with new modal predicates as is done in Epistemic Arithmetic.

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HBL-predicates 1

An L-predicate △ w.r.t. U is an HBL-predicate w.r.t. U iff it satisfies U-verifiable sentential Σ1-completeness:

◮ U ⊢ S → △S, where S is a Σ1-sentence.

We write A ✄U B for (U + A) ✄ (U + B). A sentence A is an inconsistency statement for U iff A is U-provably equivalent to △⊥ for some HBL-predicate △ w.r.t. U. Theorem: Suppose U is sequential and essentially reflexive. Then, for any sentence A, we have: ⊤ ✄U A iff A is an inconsistency for U. Proof: Using the Σ1-completeness of △ we can prove ▽A ✄U A. The proof that ⊤ ✄U △⊥ is just the proof of Feferman’s Theorem that we have seen before.

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HBL-predicates 1

An L-predicate △ w.r.t. U is an HBL-predicate w.r.t. U iff it satisfies U-verifiable sentential Σ1-completeness:

◮ U ⊢ S → △S, where S is a Σ1-sentence.

We write A ✄U B for (U + A) ✄ (U + B). A sentence A is an inconsistency statement for U iff A is U-provably equivalent to △⊥ for some HBL-predicate △ w.r.t. U. Theorem: Suppose U is sequential and essentially reflexive. Then, for any sentence A, we have: ⊤ ✄U A iff A is an inconsistency for U. Proof: Using the Σ1-completeness of △ we can prove ▽A ✄U A. The proof that ⊤ ✄U △⊥ is just the proof of Feferman’s Theorem that we have seen before.

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HBL-predicates 1

An L-predicate △ w.r.t. U is an HBL-predicate w.r.t. U iff it satisfies U-verifiable sentential Σ1-completeness:

◮ U ⊢ S → △S, where S is a Σ1-sentence.

We write A ✄U B for (U + A) ✄ (U + B). A sentence A is an inconsistency statement for U iff A is U-provably equivalent to △⊥ for some HBL-predicate △ w.r.t. U. Theorem: Suppose U is sequential and essentially reflexive. Then, for any sentence A, we have: ⊤ ✄U A iff A is an inconsistency for U. Proof: Using the Σ1-completeness of △ we can prove ▽A ✄U A. The proof that ⊤ ✄U △⊥ is just the proof of Feferman’s Theorem that we have seen before.

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HBL-predicates 1

An L-predicate △ w.r.t. U is an HBL-predicate w.r.t. U iff it satisfies U-verifiable sentential Σ1-completeness:

◮ U ⊢ S → △S, where S is a Σ1-sentence.

We write A ✄U B for (U + A) ✄ (U + B). A sentence A is an inconsistency statement for U iff A is U-provably equivalent to △⊥ for some HBL-predicate △ w.r.t. U. Theorem: Suppose U is sequential and essentially reflexive. Then, for any sentence A, we have: ⊤ ✄U A iff A is an inconsistency for U. Proof: Using the Σ1-completeness of △ we can prove ▽A ✄U A. The proof that ⊤ ✄U △⊥ is just the proof of Feferman’s Theorem that we have seen before.

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HBL-predicates 2

Conversely suppose ⊤ ✄U A. Then, by the Orey-Hájek Characterization: U ⊢ ✸nA, for all n ∈ ω. Here ✸nA is con(Un + A), where Un is axiomatized by the axioms of U with gödel number below n. We define: △B :↔ A ∨ ∃x (✷x(A → B) ∧ ✸xA). On can easily check that this predicate is HBL for U and that ⊢ A ↔ △⊥. ✷

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HBL-predicates 2

Conversely suppose ⊤ ✄U A. Then, by the Orey-Hájek Characterization: U ⊢ ✸nA, for all n ∈ ω. Here ✸nA is con(Un + A), where Un is axiomatized by the axioms of U with gödel number below n. We define: △B :↔ A ∨ ∃x (✷x(A → B) ∧ ✸xA). On can easily check that this predicate is HBL for U and that ⊢ A ↔ △⊥. ✷

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HBL-predicates 2

Conversely suppose ⊤ ✄U A. Then, by the Orey-Hájek Characterization: U ⊢ ✸nA, for all n ∈ ω. Here ✸nA is con(Un + A), where Un is axiomatized by the axioms of U with gödel number below n. We define: △B :↔ A ∨ ∃x (✷x(A → B) ∧ ✸xA). On can easily check that this predicate is HBL for U and that ⊢ A ↔ △⊥. ✷

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Consequences

It follows that there is a sentence O such that both O and ¬ O are consistency statements. So U ⊢ △0⊥ ↔ ¬ △1⊥. It follows that, for some HBL-predicate △, we have ⊤ ✄U ▽⊤. Note the contrast with ⊤ ✄ ✸N⊤. The reason that this is possible is ⊢ △A → △△A does not imply ⊢ △KA → △K△KA, but only ⊢ △KA → △K△A. in contrast we do have ⊢ ✷NKA → ✷NK✷NKA. Henk, Shavrukov & Visser constructed a far more interesting example of the phenomenon: ⊤ ✄U ▽⊤. The modal logic of that example is currently studied by Henk, de Jongh & Veltman. Combining ideas from the proof with a result of Per Lindström we find: the class of HBL-predicates for U is complete Π2.

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Consequences

It follows that there is a sentence O such that both O and ¬ O are consistency statements. So U ⊢ △0⊥ ↔ ¬ △1⊥. It follows that, for some HBL-predicate △, we have ⊤ ✄U ▽⊤. Note the contrast with ⊤ ✄ ✸N⊤. The reason that this is possible is ⊢ △A → △△A does not imply ⊢ △KA → △K△KA, but only ⊢ △KA → △K△A. in contrast we do have ⊢ ✷NKA → ✷NK✷NKA. Henk, Shavrukov & Visser constructed a far more interesting example of the phenomenon: ⊤ ✄U ▽⊤. The modal logic of that example is currently studied by Henk, de Jongh & Veltman. Combining ideas from the proof with a result of Per Lindström we find: the class of HBL-predicates for U is complete Π2.

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Consequences

It follows that there is a sentence O such that both O and ¬ O are consistency statements. So U ⊢ △0⊥ ↔ ¬ △1⊥. It follows that, for some HBL-predicate △, we have ⊤ ✄U ▽⊤. Note the contrast with ⊤ ✄ ✸N⊤. The reason that this is possible is ⊢ △A → △△A does not imply ⊢ △KA → △K△KA, but only ⊢ △KA → △K△A. in contrast we do have ⊢ ✷NKA → ✷NK✷NKA. Henk, Shavrukov & Visser constructed a far more interesting example of the phenomenon: ⊤ ✄U ▽⊤. The modal logic of that example is currently studied by Henk, de Jongh & Veltman. Combining ideas from the proof with a result of Per Lindström we find: the class of HBL-predicates for U is complete Π2.

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Consequences

It follows that there is a sentence O such that both O and ¬ O are consistency statements. So U ⊢ △0⊥ ↔ ¬ △1⊥. It follows that, for some HBL-predicate △, we have ⊤ ✄U ▽⊤. Note the contrast with ⊤ ✄ ✸N⊤. The reason that this is possible is ⊢ △A → △△A does not imply ⊢ △KA → △K△KA, but only ⊢ △KA → △K△A. in contrast we do have ⊢ ✷NKA → ✷NK✷NKA. Henk, Shavrukov & Visser constructed a far more interesting example of the phenomenon: ⊤ ✄U ▽⊤. The modal logic of that example is currently studied by Henk, de Jongh & Veltman. Combining ideas from the proof with a result of Per Lindström we find: the class of HBL-predicates for U is complete Π2.

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Consequences

It follows that there is a sentence O such that both O and ¬ O are consistency statements. So U ⊢ △0⊥ ↔ ¬ △1⊥. It follows that, for some HBL-predicate △, we have ⊤ ✄U ▽⊤. Note the contrast with ⊤ ✄ ✸N⊤. The reason that this is possible is ⊢ △A → △△A does not imply ⊢ △KA → △K△KA, but only ⊢ △KA → △K△A. in contrast we do have ⊢ ✷NKA → ✷NK✷NKA. Henk, Shavrukov & Visser constructed a far more interesting example of the phenomenon: ⊤ ✄U ▽⊤. The modal logic of that example is currently studied by Henk, de Jongh & Veltman. Combining ideas from the proof with a result of Per Lindström we find: the class of HBL-predicates for U is complete Π2.

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Overview

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

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The Logic ILM

Berarducci-Shavrukov: The following logic ILM characterizes precisely the principles of interpretability of RE sequential essentially reflexive theories.

• IL1. ⊢ ✷(A → B) → A ✄ B
• IL2. ⊢ (A ✄ B ∧ B ✄ C) → A ✄ C
• IL3. ⊢ (A ✄ C ∧ B ✄ C) → (A ∨ B) ✄ C
• IL4. ⊢ A ✄ B → (✸A → ✸B)
• IL5. ⊢ ✸A ✄ A

M ⊢ A ✄ B → (A ∧ ✷C) ✄ (A ∧ ✷C). Can we study the relation A ✄U B using the ideas introduced above?

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The Logic ILM

Berarducci-Shavrukov: The following logic ILM characterizes precisely the principles of interpretability of RE sequential essentially reflexive theories.

• IL1. ⊢ ✷(A → B) → A ✄ B
• IL2. ⊢ (A ✄ B ∧ B ✄ C) → A ✄ C
• IL3. ⊢ (A ✄ C ∧ B ✄ C) → (A ∨ B) ✄ C
• IL4. ⊢ A ✄ B → (✸A → ✸B)
• IL5. ⊢ ✸A ✄ A

M ⊢ A ✄ B → (A ∧ ✷C) ✄ (A ∧ ✷C). Can we study the relation A ✄U B using the ideas introduced above?

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The Logic ILM

Berarducci-Shavrukov: The following logic ILM characterizes precisely the principles of interpretability of RE sequential essentially reflexive theories.

• IL1. ⊢ ✷(A → B) → A ✄ B
• IL2. ⊢ (A ✄ B ∧ B ✄ C) → A ✄ C
• IL3. ⊢ (A ✄ C ∧ B ✄ C) → (A ∨ B) ✄ C
• IL4. ⊢ A ✄ B → (✸A → ✸B)
• IL5. ⊢ ✸A ✄ A

M ⊢ A ✄ B → (A ∧ ✷C) ✄ (A ∧ ✷C). Can we study the relation A ✄U B using the ideas introduced above?

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The Logic ILM

Berarducci-Shavrukov: The following logic ILM characterizes precisely the principles of interpretability of RE sequential essentially reflexive theories.

• IL1. ⊢ ✷(A → B) → A ✄ B
• IL2. ⊢ (A ✄ B ∧ B ✄ C) → A ✄ C
• IL3. ⊢ (A ✄ C ∧ B ✄ C) → (A ∨ B) ✄ C
• IL4. ⊢ A ✄ B → (✸A → ✸B)
• IL5. ⊢ ✸A ✄ A

M ⊢ A ✄ B → (A ∧ ✷C) ✄ (A ∧ ✷C). Can we study the relation A ✄U B using the ideas introduced above?

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Weak HBL-predicates

A predicate △ is a weak HBL-predicate for U iff:

• wL1. A ⊢ B

⇒ △A ⊢ △B

• wL2. ⊢ (△B ∧ △(B → C)) → △C
• wL3. ⊢ △B → △△B

wΣ1-C ⊢ (△⊤ ∧ S) → △S, where S is a Σ1-sentence (as represented in N). We can prove Löb’s Principle in the usual way.

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Weak HBL-predicates

A predicate △ is a weak HBL-predicate for U iff:

• wL1. A ⊢ B

⇒ △A ⊢ △B

• wL2. ⊢ (△B ∧ △(B → C)) → △C
• wL3. ⊢ △B → △△B

wΣ1-C ⊢ (△⊤ ∧ S) → △S, where S is a Σ1-sentence (as represented in N). We can prove Löb’s Principle in the usual way.

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Weak HBL-predicates

A predicate △ is a weak HBL-predicate for U iff:

• wL1. A ⊢ B

⇒ △A ⊢ △B

• wL2. ⊢ (△B ∧ △(B → C)) → △C
• wL3. ⊢ △B → △△B

wΣ1-C ⊢ (△⊤ ∧ S) → △S, where S is a Σ1-sentence (as represented in N). We can prove Löb’s Principle in the usual way.

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Main Theorem

Theorem: TFAE:

◮ A ✄U B ◮ for some weak HBL-predicate △ for U, we have:

A ⊢U △⊤ and △⊥ ⊢U B,

◮ for some weak HBL-predicate △ for U, we have:

U ⊢ △⊤ ↔ (A ∨ B) and U ⊢ △⊥ ↔ B. Transformation from (ii) to (iii): △∗C :↔ B ∨ (A ∧ △(B → C)). As a bonus, we gain the extra property △⊤ ⊢U △△⊥, which implies both Löb’s Principle and wL3. Terminology (?): △: a device, △⊤: in-gate, △⊥: out-gate. The relation A ⊢U △⊤ and △⊥ ⊢U B: teleportation. A beams to B?

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Main Theorem

Theorem: TFAE:

◮ A ✄U B ◮ for some weak HBL-predicate △ for U, we have:

A ⊢U △⊤ and △⊥ ⊢U B,

◮ for some weak HBL-predicate △ for U, we have:

U ⊢ △⊤ ↔ (A ∨ B) and U ⊢ △⊥ ↔ B. Transformation from (ii) to (iii): △∗C :↔ B ∨ (A ∧ △(B → C)). As a bonus, we gain the extra property △⊤ ⊢U △△⊥, which implies both Löb’s Principle and wL3. Terminology (?): △: a device, △⊤: in-gate, △⊥: out-gate. The relation A ⊢U △⊤ and △⊥ ⊢U B: teleportation. A beams to B?

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Main Theorem

Theorem: TFAE:

◮ A ✄U B ◮ for some weak HBL-predicate △ for U, we have:

A ⊢U △⊤ and △⊥ ⊢U B,

◮ for some weak HBL-predicate △ for U, we have:

U ⊢ △⊤ ↔ (A ∨ B) and U ⊢ △⊥ ↔ B. Transformation from (ii) to (iii): △∗C :↔ B ∨ (A ∧ △(B → C)). As a bonus, we gain the extra property △⊤ ⊢U △△⊥, which implies both Löb’s Principle and wL3. Terminology (?): △: a device, △⊤: in-gate, △⊥: out-gate. The relation A ⊢U △⊤ and △⊥ ⊢U B: teleportation. A beams to B?

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Main Theorem

Theorem: TFAE:

◮ A ✄U B ◮ for some weak HBL-predicate △ for U, we have:

A ⊢U △⊤ and △⊥ ⊢U B,

◮ for some weak HBL-predicate △ for U, we have:

U ⊢ △⊤ ↔ (A ∨ B) and U ⊢ △⊥ ↔ B. Transformation from (ii) to (iii): △∗C :↔ B ∨ (A ∧ △(B → C)). As a bonus, we gain the extra property △⊤ ⊢U △△⊥, which implies both Löb’s Principle and wL3. Terminology (?): △: a device, △⊤: in-gate, △⊥: out-gate. The relation A ⊢U △⊤ and △⊥ ⊢U B: teleportation. A beams to B?

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Deriving ILM from weak Löb’s Logic

Suppose we define ✄ as: for some device △ for U, we have: A ⊢U △⊤ and △⊥ ⊢U B, Can we then derive ILM purely by using the modal properties of the weak HBL-predicates? Yes we can. For example: Suppose △0 : A ✄ B and △1 : B ✄ C, then △∗ : A ✄ C, where: △∗D :↔ C ∨ (A ∧ ((¬B ∧ △0(B → ((C ∧ D) ∨ (¬C ∧ △1(C → D))))) ∨ (B ∧ △1(C → D)))). Does this yield a category with HBL-predicates as arrows? I think it should under some further conditions of ‘normalization’ of the

• predicates. I still have to do the computation for associativity of

composition.

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Deriving ILM from weak Löb’s Logic

Suppose we define ✄ as: for some device △ for U, we have: A ⊢U △⊤ and △⊥ ⊢U B, Can we then derive ILM purely by using the modal properties of the weak HBL-predicates? Yes we can. For example: Suppose △0 : A ✄ B and △1 : B ✄ C, then △∗ : A ✄ C, where: △∗D :↔ C ∨ (A ∧ ((¬B ∧ △0(B → ((C ∧ D) ∨ (¬C ∧ △1(C → D))))) ∨ (B ∧ △1(C → D)))). Does this yield a category with HBL-predicates as arrows? I think it should under some further conditions of ‘normalization’ of the

• predicates. I still have to do the computation for associativity of

composition.

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Generalizing the Semantics

If we give the same definition of teleportation but no using L-predicates we still get the logic IL. One can show that the new notion strictty extends interpretability over base theory, say, PA. I conjecture that still it does not collapse into triviality. So here is work to do. Remember that various notions of conservativity also offer alternative interpretations for ✄.

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Generalizing the Semantics

If we give the same definition of teleportation but no using L-predicates we still get the logic IL. One can show that the new notion strictty extends interpretability over base theory, say, PA. I conjecture that still it does not collapse into triviality. So here is work to do. Remember that various notions of conservativity also offer alternative interpretations for ✄.

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Generalizing the Semantics

If we give the same definition of teleportation but no using L-predicates we still get the logic IL. One can show that the new notion strictty extends interpretability over base theory, say, PA. I conjecture that still it does not collapse into triviality. So here is work to do. Remember that various notions of conservativity also offer alternative interpretations for ✄.

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Thank You