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Fixed points of self-embeddings of models of arithmetic

Saeideh Bahrami & Ali Enayat

Annals of Pure and Applied Logic 169 (6):487-513 (2018)

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Rank-initial embeddings of non-standard models of set theory.Paul Kindvall Gorbow - 2020 - Archive for Mathematical Logic 59 (5-6):517-563.details A theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then combined into a “geometric technique” used to prove several results about countable non-standard models of set theory. In particular, back-and-forth constructions are carried out to establish various generalizations and refinements of Friedman’s theorem on the existence of rank-initial embeddings between countable non-standard models of the fragment (...) Download Export citation Bookmark 3 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
Self-Embeddings of Models of Arithmetic; Fixed Points, Small Submodels, and Extendability.Saeideh Bahrami - forthcoming - Journal of Symbolic Logic:1-23.details In this paper we will show that for every cutIof any countable nonstandard model$\mathcal {M}$of$\mathrm {I}\Sigma _{1}$, eachI-small$\Sigma _{1}$-elementary submodel of$\mathcal {M}$is of the form of the set of fixed points of some proper initial self-embedding of$\mathcal {M}$iffIis a strong cut of$\mathcal {M}$. Especially, this feature will provide us with some equivalent conditions with the strongness of the standard cut in a given countable model$\mathcal {M}$of$ \mathrm {I}\Sigma _{1} $. In addition, we will find some criteria for extendability of initial (...) Download Export citation Bookmark
Tanaka’s theorem revisited.Saeideh Bahrami - 2020 - Archive for Mathematical Logic 59 (7-8):865-877.details Tanaka proved a powerful generalization of Friedman’s self-embedding theorem that states that given a countable nonstandard model \\) of the subsystem \ of second order arithmetic, and any element m of \, there is a self-embedding j of \\) onto a proper initial segment of itself such that j fixes every predecessor of m. Here we extend Tanaka’s work by establishing the following results for a countable nonstandard model \\ \)of \ and a proper cut \ of \:Theorem A. The (...) Download Export citation Bookmark

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