# Pointwise definable models of set theory

@article{Hamkins2011PointwiseDM,
title={Pointwise definable models of set theory},
author={Joel David Hamkins and David Linetsky and Jonas Reitz},
journal={The Journal of Symbolic Logic},
year={2011},
volume={78},
pages={139 - 156}
}
• Published 23 May 2011
• Mathematics, Philosophy
• The Journal of Symbolic Logic
Abstract A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V = HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is…
• Mathematics, Computer Science
Notre Dame J. Formal Log.
• 2016
The effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity is analyzed and it is shown that every (pointwise) algebraic model of ZF is actually pointwise definable.
• Computer Science
Math. Log. Q.
• 2019
The assertion that every definable set has a definable element is equivalent over ZF to the principle V=HOD , and indeed, we prove, so is the assertion merely that every Π2‐definable set has an
Abstract In this article I investigate the phenomenon of minimum and minimal models of second-order set theories, focusing on Kelley–Morse set theory KM, Gödel–Bernays set theory GB, and GB augmented
. According to the math tea argument, there must be real numbers that we cannot describe or deﬁne, because there are uncountably many real numbers, but only countably many deﬁnitions. And yet, the
• Philosophy
• 2021
It is standard in set theory to assume that Cantor’s Theorem establishes that the continuum is an uncountable set. A challenge for this position comes from the observation that through forcing one
• Mathematics, Computer Science
Notre Dame J. Formal Log.
• 2018
Ehrenfeucht’s lemma holds for the ordinal-algebraic sets, and it is proved that algebraicity and definability need not coincide in models of set theory and the internal and external notions of being ordinal algebraic need not collide.
• Environmental Science
The Journal of Symbolic Logic
• 2016
It is shown that both the definability (and, in fact, even the amenability) of the forcing relation and the truth lemma can fail for class forcing, and the forcing theorem is equivalent to the existence of a Boolean completion.
. The standard treatment of sets and deﬁnable classes in ﬁrst-order Zermelo-Fraenkel set theory accords in many respects with the Fregean foundational framework, such as the distinction between
The main theorems of this paper are (1) there is no least transitive model of Kelley--Morse set theory $\mathsf{KM}$ and (2) there is a least $\beta$-model---that is, a transitive model which is
This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the

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