A model of second-order arithmetic satisfying AC but not DC

Friedman, Sy-David; Gitman, Victoria; Kanovei, Vladimir

doi:10.1142/s0219061318500137

Cited by 23 publications

(16 citation statements)

References 8 publications

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“…As a consequence of a result of Flannagan [7,Theorem 7.1], ZF -+ GWO(R) is a conservative extension of the theory ZF -+ WO + ∀α Π 1 ∞ -DC α . Recent work of S. Friedman, Gitman, and Kanovei [10] shows that Π 1 ∞ -DC is independent of ZF -+ WO.…”

Section: (πmentioning

confidence: 99%

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Initial Self-Embeddings of Models of Set Theory

Enayat¹,

McKenzie²

2021

J. symb. log.

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By a classical theorem of Harvey Friedman (1973), every countable nonstandard model $\mathcal {M}$ of a sufficiently strong fragment of ZF has a proper rank-initial self-embedding j, i.e., j is a self-embedding of $\mathcal {M}$ such that $j[\mathcal {M}]\subsetneq \mathcal {M}$ , and the ordinal rank of each member of $j[\mathcal {M}]$ is less than the ordinal rank of each element of $\mathcal {M}\setminus j[\mathcal {M}]$ . Here, we investigate the larger family of proper initial-embeddings j of models $\mathcal {M}$ of fragments of set theory, where the image of j is a transitive submodel of $\mathcal {M}$ . Our results include the following three theorems. In what follows, $\mathrm {ZF}^-$ is $\mathrm {ZF}$ without the power set axiom; $\mathrm {WO}$ is the axiom stating that every set can be well-ordered; $\mathrm {WF}(\mathcal {M})$ is the well-founded part of $\mathcal {M}$ ; and $\Pi ^1_\infty \text{-}\mathrm {DC}_\alpha $ is the full scheme of dependent choice of length $\alpha $ .Theorem A.There is an $\omega $ -standard countable nonstandard model $\mathcal {M}$ of $\mathrm {ZF}^-+\mathrm {WO}$ that carries no initial self-embedding $j:\mathcal {M} \longrightarrow \mathcal {M}$ other than the identity embedding.Theorem B.Every countable $\omega $ -nonstandard model $\mathcal {M}$ of $\ \mathrm {ZF}$ is isomorphic to a transitive submodel of the hereditarily countable sets of its own constructible universe $L^{\mathcal {M}}$ .Theorem C.The following three conditions are equivalent for a countable nonstandard model $\mathcal {M}$ of $\mathrm {ZF}^{-}+\mathrm {WO}+\forall \alpha \ \Pi ^1_\infty \text{-}\mathrm {DC}_\alpha $ . (I)There is a cardinal in $\mathcal {M}$ that is a strict upper bound for the cardinality of each member of $\mathrm {WF}(\mathcal {M})$ .(II) $\mathrm {WF}(\mathcal {M})$ satisfies the powerset axiom.(III)For all $n \in \omega $ and for all $b \in M$ , there exists a proper initial self-embedding $j: \mathcal {M} \longrightarrow \mathcal {M}$ such that $b \in \mathrm {rng}(j)$ and $j[\mathcal {M}] \prec _n \mathcal {M}$ .

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Section: (πmentioning

confidence: 99%

“…We now turn to applying Theorem 5.3 to finding transitive partially elementary substructures of nonstandard models of ZF -+ WO. Despite the failure of reflection in ZF -+ WO [10], Quinsey [23,Corollary 6.9] employed indicators and methods from infinitary logic to show the following: Theorem 5.17 (Quinsey). Let n ∈ .…”

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confidence: 99%

Initial Self-Embeddings of Models of Set Theory

Enayat¹,

McKenzie²

2021

J. symb. log.

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“…Meanwhile, KP is not reflexive, nor is any finitely axiomatizable theory. In general, the reflexive property is weaker than the reflection theorem for T , since models of , for example, do not necessarily satisfy the reflection theorem, as proved in [4], but nevertheless, the theory is reflexive to transitive sets, since if and , then we can code a with a set of ordinals and consider , which is a model of containing a and having an -hierarchy, which is enough to prove the reflection theorem. So we get transitive models of the form satisfying any desired finite fragment of and containing the set a .…”

Section: A Refined Version Of the Main Theorem With Applicationsmentioning

confidence: 99%

THE Σ₁-DEFINABLE UNIVERSAL FINITE SEQUENCE

Hamkins¹,

Williams²

2021

J. symb. log.

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We introduce the $\Sigma _1$ -definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, (i) the sequence is $\Sigma _1$ -definable and provably finite; (ii) the sequence is empty in transitive models; and (iii) if M is a countable model of set theory in which the sequence is s and t is any finite extension of s in this model, then there is an end-extension of M to a model in which the sequence is t. Our proof method grows out of a new infinitary-logic-free proof of the Barwise extension theorem, by which any countable model of set theory is end-extended to a model of $V=L$ or indeed any theory true in a suitable submodel of the original model. The main theorem settles the modal logic of end-extensional potentialism, showing that the potentialist validities of the models of set theory under end-extensions are exactly the assertions of S4. Finally, we introduce the end-extensional maximality principle, which asserts that every possibly necessary sentence is already true, and show that every countable model extends to a model satisfying it.

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“…4. It will be interesting to apply the hidden invariance technique to some other forcing notions and coding systems (those not of the almost-disjoint type), such as in [21,22].…”

Section: Approximations Of the N-complete Forcing Notionmentioning

confidence: 99%