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Page 1: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Metamathematical Theorems with

regard to Gödel Numberings

Balthasar Grabmayr

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

Page 2: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

1 Invariance of Gödel's Second Theorem

2 Acceptable Numberings

3 Proof of the Invariance Claim

Page 3: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

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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Page 4: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

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

?

1 / 14

Page 5: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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.

2 / 14

Page 6: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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.

2 / 14

Page 7: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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.

3 / 14

Page 8: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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.

3 / 14

Page 9: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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.

3 / 14

Page 10: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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

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).

here's the hidden text

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Page 11: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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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Page 12: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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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Page 13: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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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Page 14: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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:

E k E

G k G

αk

σE

α

σG

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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Page 15: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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:

E k E

G k G

αk

σE

α

σG

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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Page 16: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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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Page 17: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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].

8 / 14

Page 18: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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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Page 19: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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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Page 20: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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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Page 21: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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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Page 22: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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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Page 23: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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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Page 24: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

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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Page 25: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

Summary

Initial Question

How to justify the (meta-theoretical) inference G2IG2

?

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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Page 26: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

Summary

Initial Question

How to justify the (meta-theoretical) inference G2IG2

?

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

14 / 14

Page 27: Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with regard to Gödel Numberings Balthasar Grabmayr PhDs in Logic IX (Bochum) 2nd of Ma,y

Invariance of Gödel's Second TheoremAcceptable Numberings

Proof of the Invariance Claim

Summary

Initial Question

How to justify the (meta-theoretical) inference G2IG2

?

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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