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End-extensions preserving power set

Thomas Forster & Richard Kaye

Journal of Symbolic Logic 56 (1):323-328 (1991)

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On the relative strengths of fragments of collection.Zachiri McKenzie - 2019 - Mathematical Logic Quarterly 65 (1):80-94.details Let be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, Δ0‐separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set‐theoretic collection scheme to. We focus on two common parameterisations of the collection: ‐collection, which is the usual collection scheme restricted to ‐formulae, and strong ‐collection, which is equivalent to ‐collection plus ‐separation. The main result of this paper shows that for (...) Download Export citation Bookmark 2 citations
Automorphisms of models of set theory and extensions of NFU.Zachiri McKenzie - 2015 - Annals of Pure and Applied Logic 166 (5):601-638.details Download Export citation Bookmark 5 citations
The strength of Mac Lane set theory.A. R. D. Mathias - 2001 - Annals of Pure and Applied Logic 110 (1-3):107-234.details Saunders Mac Lane has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, , of Transitive Containment, we shall refer as . His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasises, one that is (...) Download Export citation Bookmark 37 citations
Initial self-embeddings of models of set theory.Ali Enayat & Zachiri Mckenzie - 2021 - Journal of Symbolic Logic 86 (4):1584-1611.details By a classical theorem of Harvey Friedman, 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 (...) Download Export citation Bookmark
Largest initial segments pointwise fixed by automorphisms of models of set theory.Ali Enayat, Matt Kaufmann & Zachiri McKenzie - 2018 - Archive for Mathematical Logic 57 (1-2):91-139.details Given a model \ of set theory, and a nontrivial automorphism j of \, let \\) be the submodel of \ whose universe consists of elements m of \ such that \=x\) for every x in the transitive closure of m ). Here we study the class \ of structures of the form \\), where the ambient model \ satisfies a frugal yet robust fragment of \ known as \, and \=m\) whenever m is a finite ordinal in the sense (...) Download Export citation Bookmark 3 citations

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