  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
$\text{ZFC}+V=L[\mu]$.
