invariance of metamathematical theorems with …...invariance of metamathematical theorems with...

Invariance of Metamathematical Theorems with

regard to Gödel Numberings

Balthasar Grabmayr

PhDs in Logic IX (Bochum)2nd of May, 2017

1 Invariance of Gödel's Second Theorem

2 Acceptable Numberings

3 Proof of the Invariance Claim

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

The Philosophical Reading of Gödel's Second Theorem

Let T ⊇ EA be a consistent, r.e. L-theory and σ(L) = 0, S,+, ·.

Gödel's Second Theorem

(G2) T 6` ConT

Philosophical Interpretation of Gödel's Second Theorem

(IG2) T does not prove its consistency.

Question

How to justify the (meta-theoretical) inference G2IG2

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The Philosophical Reading of Gödel's Second Theorem

Let T ⊇ EA be a consistent, r.e. L-theory and σ(L) = 0, S,+, ·.

Gödel's Second Theorem

(G2) T 6` ConT

Philosophical Interpretation of Gödel's Second Theorem

(IG2) T does not prove its consistency.

Question

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Generalising Gödel's Second Theorem

Theorem (Hilbert, Bernays, Löb)

T 6` ¬Pr(p0 = 1q), for all formulæ Pr(x) satisfying Löb(T ).

Denition

A formula Pr(x) satises Löb's conditions (for T ), in short Löb(T ),if for all sentences φ and ψ:

Löb1(T ) T ` φ implies T ` Pr(pφq);

Löb2(T ) T ` Pr(pφq) ∧ Pr(pφ→ ψq)→ Pr(pψq);

Löb3(T ) T ` Pr(pφq)→ Pr(pPr(pφq)q).

Problem: Dependence on a specic Gödel numbering.

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Generalising Gödel's Second Theorem

Theorem (Hilbert, Bernays, Löb)

T 6` ¬Pr(p0 = 1q), for all formulæ Pr(x) satisfying Löb(T ).

Denition

A formula Pr(x) satises Löb's conditions (for T ), in short Löb(T ),if for all sentences φ and ψ:

Löb1(T ) T ` φ implies T ` Pr(pφq);

Löb2(T ) T ` Pr(pφq) ∧ Pr(pφ→ ψq)→ Pr(pψq);

Löb3(T ) T ` Pr(pφq)→ Pr(pPr(pφq)q).

Problem: Dependence on a specic Gödel numbering.

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Löb's Conditions Relativised to Numberings

Theorem

T 6` ¬Prα(p0 = 1qα), for all numberings α and for all formulæ

Prα(x) satisfying Löb(T , α).

Denition

A formula Pr(x) satises Löb's conditions (for T ) relative to a

numbering α, in short Löb(T , α), if for all sentences φ and ψ:

Löb1(T , α) T ` φ implies T ` Pr(pφqα);

Löb2(T , α) T ` Pr(pφqα) ∧ Pr(pφ→ ψqα)→ Pr(pψqα);

Löb3(T , α) T ` Pr(pφqα)→ Pr(pPr(pφqα)qα).

Problem: Deviant Gödel numberings.

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Theorem

Denition

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Theorem

Denition

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Invariance of G2 with regard to Numberings

Invariance Theorem

T 6` ¬Prα(p0 = 1qα), for all acceptable numberings α and for

all formulæ Prα(x) satisfying Löb(T , α).

Denition

here's the hidden text

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Roadmap: Acceptable Numberings

Idea: Every acceptable numbering is computable.

Step 1: General framework to model expressions: Equationaltheories and initial algebra semantics(only constraint: nite alphabet)

Step 2: Conceptually adequate denition of computabilityover arbitrary models of expressions

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Modelling Expressions

Let Ω = Ω+∪Ω0 be a nite algebraic signature, with constructorsΩ+ and generators Ω0 (only containing constant symbols).Let E be a set of Ω+-equations over a (disjoint) set of variables.

Denition

We take universes of expressions to be initial algebras E inMod(Ω, E). If E contains all L-terms and L-formulæ, we call E auniverse of arithmetical expressions.

Example

Set of nite sequences over a nite arithmetical alphabet togetherwith a concatenation operation

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Modelling Expressions

Let Ω = Ω+∪Ω0 be a nite algebraic signature, with constructorsΩ+ and generators Ω0 (only containing constant symbols).Let E be a set of Ω+-equations over a (disjoint) set of variables.

Denition

We take universes of expressions to be initial algebras E inMod(Ω, E). If E contains all L-terms and L-formulæ, we call E auniverse of arithmetical expressions.

Example

Set of nite sequences over a nite arithmetical alphabet togetherwith a concatenation operation

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Turning Gödel Numbers into an Algebra

Let E be a universe of expressions and α : E → N an injectivefunction. Set G = α(E ).Dene for each σ ∈ Ω+

k a tracking function σG : G k → G of σE(under α) such that the diagram commutes:

G together with the tracking functions forms an Ω+-algebra G.

α : E→ G is an Ω+-isomorphism.

G is Ω+-generated over α(sE) | s ∈ Ω0.

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Turning Gödel Numbers into an Algebra

Let E be a universe of expressions and α : E → N an injectivefunction. Set G = α(E ).Dene for each σ ∈ Ω+

k a tracking function σG : G k → G of σE(under α) such that the diagram commutes:

G together with the tracking functions forms an Ω+-algebra G.

α : E→ G is an Ω+-isomorphism.

G is Ω+-generated over α(sE) | s ∈ Ω0.

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Dening Computability of Numberings

Denition

Let E be a universe of expressions. We call a numbering α : E → Ncomputable, if

1 α(E ) is decidable;

2 for each σ ∈ Ω+k there exists a recursive tracking function

σN : α(E )k → α(E ) of σE, i.e. for all t1, . . . , tk ∈ E

σN(α(t1), . . . , α(tk)) = α(σE(t1, . . . , tk)).

For the adequacy of this denition see e.g. [U. Boker and N.Dershowitz, 2008].

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Dening Computability of Numberings

Denition

Let E be a universe of expressions. We call a numbering α : E → Ncomputable, if

1 α(E ) is decidable;

2 for each σ ∈ Ω+k there exists a recursive tracking function

σN : α(E )k → α(E ) of σE, i.e. for all t1, . . . , tk ∈ E

σN(α(t1), . . . , α(tk)) = α(σE(t1, . . . , tk)).

For the adequacy of this denition see e.g. [U. Boker and N.Dershowitz, 2008].

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Computable Numberings are Computably Equivalent

Denition

We call two numberings α, β of a set (computably) equivalent(write α ∼ β), if the functions α β−1 : β(S)→ N andβ α−1 : α(S)→ N are recursive (in particular, α(S) and β(S) aredecidable).

Theorem (Malcev)

Let E be a universe of expressions and let α and β be computable

numberings of E. Then α ∼ β.

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Invariance of Tarski's Theorem

Corollary

The decidable, recursive enumerable and arithmetical subsets of a

universe of expressions are invariant with regard to computable

numberings.

Corollary

Tarski's Theorem is invariant with regard to computable

numberings.

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Formalising Computable Equivalence

Theorem

Let α and β be computable numberings of a set of arithmetical

expressions E and let T ⊇ Q. Then there is a binumeration

f (x , y) ∈ FmlL of β α−1 such that for each formula

Prα(x) ∈ FmlL which satises Löb(T , α) there exists a formula

Prβ(x) ∈ FmlL which satises Löb(T , β), such that

Q ` ∀x , y(f (x , y)→ (Prα(x)↔ Prβ(y))).

Furthermore, if Prα(x) numerates α(T`), then Prβ(x) numerates

β(T`).

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Invariance of G2 Regarding Acceptable Numberings

Corollary

For all computable numberings α, consistent, r.e. theories T ⊇ EAand arithmetical formulæ Prα(x) satisfying Löb(T , α), it holds that

T 6` ¬Prα(p0 = 1qα).

Proof.

Let γ be a standard numbering. Then α ∼ γ, hence there is abinumeration f (x , y) of γ α−1 and a formula Prγ(x) satisfyingLöb1-3 (for T ) relative to γ such that

Q ` ∀x , y(f (x , y)→ (Prα(x)↔ Prγ(y))).

Therefore Q ` ¬Prα(p0 = 1qα)↔ ¬Prγ(p0 = 1qγ).But T 6` ¬Prγ(p0 = 1qγ).

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Invariance of G2 Regarding Acceptable Numberings

Corollary

For all computable numberings α, consistent, r.e. theories T ⊇ EAand arithmetical formulæ Prα(x) satisfying Löb(T , α), it holds that

T 6` ¬Prα(p0 = 1qα).

Proof.

Let γ be a standard numbering. Then α ∼ γ, hence there is abinumeration f (x , y) of γ α−1 and a formula Prγ(x) satisfyingLöb1-3 (for T ) relative to γ such that

Q ` ∀x , y(f (x , y)→ (Prα(x)↔ Prγ(y))).

Therefore Q ` ¬Prα(p0 = 1qα)↔ ¬Prγ(p0 = 1qγ).But T 6` ¬Prγ(p0 = 1qγ).

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A Concluding Remark

Do non-standard numberings even allow the construction of a(non-trivial) provability predicate satisfying Löb's derivabilityconditions?

Yes, see:

Corollary

For all acceptable numberings α and consistent, recursively

enumerable theories T ⊇ EA, there exists a formula PrαT (x) which

satises Löb(T , α) and numerates α(φ) | T ` φ in EA.

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A Concluding Remark

Do non-standard numberings even allow the construction of a(non-trivial) provability predicate satisfying Löb's derivabilityconditions? Yes, see:

Corollary

For all acceptable numberings α and consistent, recursively

enumerable theories T ⊇ EA, there exists a formula PrαT (x) which

satises Löb(T , α) and numerates α(φ) | T ` φ in EA.

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Summary

Initial Question

1 Formulation of a numbering-sensitive version of Gödel'sSecond Theorem

General framework for modelling expressions

Conceptually clear notion of computability for numberings

Computability as necessary condition for acceptability

2 Proof of Invariance of Gödel's Second Theorem regardingacceptable Numberings

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Summary

Initial Question

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Summary

Initial Question

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invariance of metamathematical theorems with …...invariance of metamathematical theorems with...

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