# Pointwise definable models of set theory

@article{Hamkins2013PointwiseDM, title={Pointwise definable models of set theory}, author={Joel David Hamkins and David Linetsky and Jonas Reitz}, journal={The Journal of Symbolic Logic}, year={2013}, volume={78}, pages={139 - 156} }

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…

## 22 Citations

### Algebraicity and Implicit Definability in Set Theory

- Mathematics, Computer ScienceNotre 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.

### When does every definable nonempty set have a definable element?

- Computer ScienceMath. 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…

### MINIMUM MODELS OF SECOND-ORDER SET THEORIES

- MathematicsThe Journal of Symbolic Logic
- 2019

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…

### Every countable model of arithmetic or set theory has a pointwise-definable end extension

- Mathematics, Philosophy
- 2022

. 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…

### Ehrenfeucht's Lemma in Set Theory

- Mathematics, Computer ScienceNotre 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.

### CLASS FORCING, THE FORCING THEOREM AND BOOLEAN COMPLETIONS

- Environmental ScienceThe 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.

### Least models of second-order set theories

- Mathematics
- 2017

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…

### The Structure of Models of Second-order Set Theories

- Mathematics
- 2018

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…

### Categorical large cardinals and the tension between categoricity and set-theoretic reflection

- Philosophy, Mathematics
- 2020

Inspired by Zermelo's quasi-categoricity result characterizing the models of second-order Zermelo-Fraenkel set theory $\text{ZFC}_2$, we investigate when those models are fully categorical,…

### THE SET-THEORETIC MULTIVERSE

- PhilosophyThe Review of Symbolic Logic
- 2012

It is argued that the continuum hypothesis is settled on the multiverse view by the authors' extensive knowledge about how it behaves in theMultiverse, and as a result it can no longer be settled in the manner formerly hoped for.

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