# 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} }

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…

## 23 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…

### Countabilism and Maximality Principles

- 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…

### 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.

### Fregean abstraction in Zermelo-Fraenkel set theory: a deflationary account

- Philosophy
- 2022

. 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…

### 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…

## References

SHOWING 1-10 OF 19 REFERENCES

### Models of set theory with definable ordinals

- MathematicsArch. Math. Log.
- 2005

This work provides a comprehensive treatment of Paris models and shows that for every infinite cardinal κ there is a Paris model of ZF of cardinality κ which has a nontrivial automorphism.

### Forcing and models of arithmetic

- Mathematics
- 1974

It is shown that every countable model of Peano arithmetic with finitely many extra predicates (or of ZFC with finitely many extra predicates) is a reduct of a pointwise definable such model. This…

### The ground axiom

- Philosophy, EconomicsJournal of Symbolic Logic
- 2007

The Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model, is first-order expressible, and any model of ZFC has a class forcing extension which satisfies it.

### Counting models of set theory

- Mathematics
- 2002

Let T denote a completion of ZF. We are interested in the number μ(T ) of isomorphism types of countable well-founded models of T . Given any countable order type τ , we are also interested in the…

### The Structure of Models of Peano Arithmetic

- Mathematics
- 2006

Preface 1. Basics 2. Extensions 3. Minimal and other types 4. Substructure lattices 5. How to control types 6. Generics and forcing 7. Cuts 8. Automorphisms of recursively saturated models 9.…

### ON RESURRECTION AXIOMS

- PhilosophyThe Journal of Symbolic Logic
- 2015

A stronger form of resurrection is introduced and it is shown that it gives rise to families of axioms which are consistent relative to extendible cardinals, and which imply the strongest known instances of forcing axiomatic, such as Martin’s Maximum++.

### Introduction to mathematical logic

- PhilosophyUniversitext
- 1973

This book discusses the semantics of Predicate Logic, and some of theorems of A. Robinson, Craig and Beth's treatment of Peano's Axiom System.

### An Introduction to Mathematical Logic

- Philosophy
- 1995

Background semantics of propositional logic propositional logic first order languages first order logic logic and mathematics Godel's "Incompleteness Theorem" ,using Church's thesis recursive…

### Archive for mathematical logic

- Philosophy
- 1987

The journal publishes research papers and occasionally surveys or expositions on mathematical logic. Contributions are also welcomed from other related areas, such as theoretical computer science or…