Marginalia on a Theorem of Woodin

Blanck, Rasmus; Enayat, Ali

doi:10.1017/jsl.2016.8

Cited by 6 publications

(11 citation statements)

References 19 publications

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“…Proof sketch. It is straightforward to adapt the construction in the proof of Theorem 4.31 to produce an M-finite binary sequence S e , rather than a set [4,18,48]. Assume an enumeration of Σ m+2 formulae in which every Σ m+2 formula occurs infinitely often, and that every finite binary sequence codes such a formula.…”

Section: Model Theory Of Arithmeticmentioning

confidence: 99%

“…Moreover, IΣ n+1 ∀x(Pr (x) ↔ Pr (x)). 4 Bibliographical remark. Craig's [7] formulation pertains to r.e.…”

mentioning

confidence: 99%

“…A similar remark applies also to many of the following facts, and to the statements of some of the theorems in Section 4. 4 For n = 0, it is known that IΣ 1 suffices to show that the Craigified theory is deductively equivalent to the original one [17,remark III.2.30]. By contrast, this is not true of IΣ 0 + exp [47].…”

mentioning

confidence: 99%

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Hierarchical Incompleteness Results for Arithmetically Definable Extensions of Fragments of Arithmetic

Blanck¹

2021

The Review of Symbolic Logic

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There has been a recent interest in hierarchical generalizations of classic incompleteness results. This paper provides evidence that such generalizations are readily obtainable from suitably formulated hierarchical versions of the principles used in the original proofs. By collecting such principles, we prove hierarchical versions of Mostowski’s theorem on independent formulae, Kripke’s theorem on flexible formulae, Woodin’s theorem on the universal algorithm, and a few related results. As a corollary, we obtain the expected result that the formula expressing “ $\mathrm {T}$ is $\Sigma _n$ -ill” is a canonical example of a $\Sigma _{n+1}$ formula that is $\Pi _{n+1}$ -conservative over $\mathrm {T}$ .

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Section: Model Theory Of Arithmeticmentioning

confidence: 99%

“…Moreover, IΣ n+1 ∀x(Pr (x) ↔ Pr (x)). 4 Bibliographical remark. Craig's [7] formulation pertains to r.e.…”

mentioning

confidence: 99%

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Hierarchical Incompleteness Results for Arithmetically Definable Extensions of Fragments of Arithmetic

Blanck¹

2021

The Review of Symbolic Logic

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“…Of course, Σ 1 -soundness is a strengthening of Con(PA + ), and in any model of ¬ Con(PA + ) the algorithm will find proofs and therefore have a successful stage. Blanck and Enayat [BE17] prove that their version of the universal algorithm, as well as Woodin's original algorithm, has the property that it enumerates a nonempty sequence if and only if ¬ Con(PA + ). It seems that the argument can be made to work also for the algorithm here.…”

Section: The Universal Algorithmmentioning

confidence: 99%

The Modal Logic of Set-Theoretic Potentialism and the Potentialist Maximality Principles

Hamkins¹,

Linnebo²

2019

The Review of Symbolic Logic

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I investigate the modal commitments of various conceptions of the philosophy of arithmetic potentialism. Specifically, I consider the natural potentialist systems arising from the models of arithmetic under their natural extension concepts, such as end-extensions, arbitrary extensions, conservative extensions and more. In these potentialist systems, I show, the propositional modal assertions that are valid with respect to all arithmetic assertions with parameters are exactly the assertions of S4. With respect to sentences, however, the validities of a model lie between S4 and S5, and these bounds are sharp in that there are models realizing both endpoints. For a model of arithmetic to validate S5 is precisely to fulfill the arithmetic maximality principle, which asserts that every possibly necessary statement is already true, and these models are equivalently characterized as those satisfying a maximal Σ 1 theory. The main S4 analysis makes fundamental use of the universal algorithm, of which this article provides a simplified, self-contained account. The paper concludes with a discussion of how the philosophical differences of several fundamentally different potentialist attitudes-linear inevitability, convergent potentialism and radical branching possibility-are expressed by their corresponding potentialist modal validities.

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“…The equivalence of ( 1) and ( 2) for finitely axiomatisable theories seems to have been known to experts for some time, while the equivalence of ( 2) and (3) for r.e. fragments of PA is found in [3,Theorem 2.11]. The generalisation to Σ n+1 -definable theories present no further difficulties.…”

Section: Let Pr Kmentioning

confidence: 99%

Hierarchical incompleteness results for arithmetically definable extensions of fragments of arithmetic

Blanck

2018

Preprint

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There has been a recent interest in hierarchical generalisations of classic incompleteness results. This paper provides evidence that such generalisations are readibly obtainable from suitably hierarchical versions of the principles used in the original proof. By collecting such principles, we prove hierarchical versions of Mostowski's theorem on independent formulae, Kripke's theorem on flexible formulae, and a number of further generalisations thereof. As a corollary, we obtain the expected result that the formula expressing "T is Σn-ill" is a canonical example of a Σn+1 formula that is Πn+1-conservative over T. * This paper is based on a cross section of the author's doctoral thesis [2]: in particular, the Corollary to Theorem 3 (with a different proof), and what is essentially Theorem 4 have already appeared there. I am grateful to Ali Enayat, Joel Hamkins, and Volodya Shavrukov for inspiration, discussion, and guidance.

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Marginalia on a Theorem of Woodin

Cited by 6 publications

References 19 publications

Hierarchical Incompleteness Results for Arithmetically Definable Extensions of Fragments of Arithmetic

Hierarchical Incompleteness Results for Arithmetically Definable Extensions of Fragments of Arithmetic

The Modal Logic of Set-Theoretic Potentialism and the Potentialist Maximality Principles

Hierarchical incompleteness results for arithmetically definable extensions of fragments of arithmetic

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