# 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}
}
• 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…

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## References

SHOWING 1-10 OF 19 REFERENCES

### Models of set theory with definable ordinals

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

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

• J. Reitz
• Philosophy, Economics
Journal 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

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

### Large cardinals and definable well-orders on the universe

A reverse Easton forcing iteration is used to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals, by choosing the cardinals at which coding occurs sufficiently sparsely.

### Set-theoretic geology

• Mathematics
Ann. Pure Appl. Log.
• 2015

### 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

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

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

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