Disjunctions With Stopping Conditions

Kossak, Roman; Wcisło, Bartosz

doi:10.1017/bsl.2019.55

Cited by 9 publications

(14 citation statements)

References 8 publications

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“…The original proof relied on the M -provability methods. A closely related result obtained via the Enayat-Visser construction (and thus possibly easier to digest) can be found in [12].…”

Section: Corollary 24 If M Is Any Model Of Pa (Of An Arbitrary Cardin...mentioning

confidence: 67%

“…To prove the "moreover" part, notice that we can consider sentences of the form φ a defined as a = a ∧ φ. Again, for any n ∈ ω, there exists a pair of sentences φ a , ψ a ∈ Sent L PA (M ) such that they have the same n-types and 12 This Proposition essentially appears in the second proof of Theorem 1 in [7], attributed to Woodin, where it is stated and proved in the context of recursively saturated satisfaction classes, rather than inductive ones.…”

Section: Satisfaction Classes and Automorphismsmentioning

confidence: 92%

“…L PA (M ) such that tp(φ, a) = tp(ψ, a), but (φ, ∅) ∈ S, (ψ, ∅) / ∈ S. 12 In the proof below, by an n-type of an element a, we mean the set of arithmetical formulae in a single variable of syntactic depth n satisfied by a. We denote the n-type of a by tp n (a).…”

Section: Satisfaction Classes and Automorphismsmentioning

confidence: 99%

“…Actually, the regularity requirement is not needed in Theorem 29 and consequently in Theorem 27. However, the cited result is formulated in [12] in the context of CT − , where we automatically assume regularity, so we decided to leave it as is.…”

Section: Satisfaction Classes and Automorphismsmentioning

confidence: 99%

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Full satisfaction classes, definability, and automorphisms

Wcisło¹

2021

Preprint

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We show that for every countable recursively saturated model M of Peano Arithmetic and every subset A ⊆ M , there exists a full satisfaction class S A ⊂ M 2 such that A is definable in (M, S A ) without parametres. It follows that in every such model, there exists a full satisfaction class which makes every element definable and thus the expanded model is minimal and rigid. On the other hand, we show that for every full satisfaction class S there are two elements which have the same arithmetical type, but exactly one of them is in S. In particular, the automorphism group of a model expanded with a satisfaction class is never equal to the automorphism group of the original model. The analogue of many of the results proved here for full satisfaction classes were obtained by Roman Kossak for partial inductive satisfaction classes. However, most of the proofs relied heavily on the induction scheme in a crucial way, so recapturing the results in the setting of full satisfaction classes requires quite different arguments.

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“…The original proof relied on the M -provability methods. A closely related result obtained via the Enayat-Visser construction (and thus possibly easier to digest) can be found in [12].…”

Section: Corollary 24 If M Is Any Model Of Pa (Of An Arbitrary Cardin...mentioning

confidence: 67%

Section: Satisfaction Classes and Automorphismsmentioning

confidence: 92%

Section: Satisfaction Classes and Automorphismsmentioning

confidence: 99%

Section: Satisfaction Classes and Automorphismsmentioning

confidence: 99%

See 2 more Smart Citations

Full satisfaction classes, definability, and automorphisms

Wcisło¹

2021

Preprint

Self Cite

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“…• T ⊂ T ′ , in which T ′ cannot be further extended to a predicate T ′′ satisfying CT − . The proof of this fact will appear in [14].…”

Section: ∀X∃yφ(x Y)mentioning

confidence: 90%

Local collection scheme and end-extensions of models of compositional truth

Łełyk,

Wcisło

2020

Preprint

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We introduce a principle of local collection for compositional truth predicates and show that it is conservative over the classically compositional theory of truth in the arithmetical setting. This axiom states that upon restriction to formulae of any syntactic complexity, the resulting predicate satisfies full collection. In particular, arguments using collection for the truth predicate applied to sentences occurring in any given (code of a) proof do not suffice to show that the conclusion of that proof is true, in stark contrast to the case of induction scheme.We analyse various further results concerning end-extensions of models of compositional truth and collection scheme for the compositional truth predicate.

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Compositional truth with propositional tautologies and quantifier-free correctness

Wcisło

2023

Arch. Math. Logic

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In Cieśliński (J Philos Logic 39:325–337, 2010), Cieśliński asked whether compositional truth theory with the additional axiom that all propositional tautologies are true is conservative over Peano Arithmetic. We provide a partial answer to this question, showing that if we additionally assume that truth predicate agrees with arithmetical truth on quantifier-free sentences, the resulting theory is as strong as $$\Delta _0$$ Δ 0 -induction for the compositional truth predicate, hence non-conservative. On the other hand, it can be shown with a routine argument that the principle of quantifier-free correctness is itself conservative.

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Disjunctions With Stopping Conditions

Cited by 9 publications

References 8 publications

Full satisfaction classes, definability, and automorphisms

Full satisfaction classes, definability, and automorphisms

Local collection scheme and end-extensions of models of compositional truth

Compositional truth with propositional tautologies and quantifier-free correctness

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