Invariance of Metamathematical Theorems with …...Invariance of Metamathematical Theorems with...
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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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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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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.
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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.
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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.
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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.
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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.
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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