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

TL;DR

The paper demonstrates that every countable model of arithmetic or set theory can be extended to a pointwise-definable model, where each element is uniquely definable without parameters, challenging traditional views on definability.

Contribution

Introduces a new flexible method for constructing pointwise-definable models of arithmetic and set theory, applicable to all countable models of ZF and PA.

Findings

Every countable model of ZF has a pointwise-definable end extension.

Every countable model of PA has a pointwise-definable end extension.

Extensions can satisfy additional inner model properties like $V=L$ or $V=L[ ext{μ}]$.

Abstract

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 existence of pointwise-definable models of set theory, in which every individual is definable without parameters, challenges this conclusion. In this article, I introduce a flexible new method for constructing pointwise-definable models of arithmetic and set theory, showing furthermore that every countable model of Zermelo-Fraenkel ZF set theory and of Peano arithmetic PA has a pointwise-definable end extension. In the arithmetic case, I use the universal algorithm and its $\Sigma_n$ generalizations to build a progressively elementary tower making any desired individual $a_n$ definable at each stage $n$, while preserving these definitions through to the limit model, which can thus be arranged to be pointwise definable. A similar method works in set theory, and one can moreover achieve $V=L$ in the extension or indeed any other suitable theory holding in an inner model of the original model, thereby fulfilling the resurrection phenomenon. For example, every countable model of ZF with an inner model with a measurable cardinal has an end extension to a pointwise-definable model of $\text{ZFC}+V=L[\mu]$.