Initial self-embeddings of models of set theory

Journal of Symbolic Logic 86 (4):1584-1611 (2021)
  Copy   BIBTEX

Abstract

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 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; $\mathrm {WF}$ 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 $. There is a cardinal in $\mathcal {M}$ that is a strict upper bound for the cardinality of each member of $\mathrm {WF}$. $\mathrm {WF}$ satisfies the powerset axiom.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 $b \in \mathrm {rng}$ and $j[\mathcal {M}] \prec _n \mathcal {M}$.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 91,322

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Rank-initial embeddings of non-standard models of set theory.Paul Kindvall Gorbow - 2020 - Archive for Mathematical Logic 59 (5-6):517-563.
Submodels of Kripke models.Albert Visser - 2001 - Archive for Mathematical Logic 40 (4):277-295.
Co-critical points of elementary embeddings.Michael Sheard - 1985 - Journal of Symbolic Logic 50 (1):220-226.
Condensable models of set theory.Ali Enayat - 2022 - Archive for Mathematical Logic 61 (3):299-315.
Tanaka’s theorem revisited.Saeideh Bahrami - 2020 - Archive for Mathematical Logic 59 (7-8):865-877.
An example related to Gregory’s Theorem.J. Johnson, J. F. Knight, V. Ocasio & S. VanDenDriessche - 2013 - Archive for Mathematical Logic 52 (3-4):419-434.
Universal Structures.Saharon Shelah - 2017 - Notre Dame Journal of Formal Logic 58 (2):159-177.
Fixed points of self-embeddings of models of arithmetic.Saeideh Bahrami & Ali Enayat - 2018 - Annals of Pure and Applied Logic 169 (6):487-513.
LD-Algebras Beyond I0.Vincenzo Dimonte - 2019 - Notre Dame Journal of Formal Logic 60 (3):395-405.
Leibnizian models of set theory.Ali Enayat - 2004 - Journal of Symbolic Logic 69 (3):775-789.
Guessing models and generalized Laver diamond.Matteo Viale - 2012 - Annals of Pure and Applied Logic 163 (11):1660-1678.
Why Can Computers Understand Natural Language?Juan Luis Gastaldi - 2020 - Philosophy and Technology 34 (1):149-214.

Analytics

Added to PP
2022-04-08

Downloads
5 (#1,498,791)

6 months
4 (#797,377)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Citations of this work

No citations found.

Add more citations

References found in this work

The strength of Mac Lane set theory.A. R. D. Mathias - 2001 - Annals of Pure and Applied Logic 110 (1-3):107-234.
Set Theory.Thomas Jech - 1999 - Studia Logica 63 (2):300-300.
On Interpretations of Arithmetic and Set Theory.Richard Kaye & Tin Lok Wong - 2007 - Notre Dame Journal of Formal Logic 48 (4):497-510.
What is the theory without power set?Victoria Gitman, Joel David Hamkins & Thomas A. Johnstone - 2016 - Mathematical Logic Quarterly 62 (4-5):391-406.
Fixed points of self-embeddings of models of arithmetic.Saeideh Bahrami & Ali Enayat - 2018 - Annals of Pure and Applied Logic 169 (6):487-513.

View all 12 references / Add more references