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According to the math tea argument, there must be real numbers that we cannotdescribe or define, because there are uncountably many real numbers, but onlycountably many definitions. And yet, the existence of pointwise definablemodels of set theory, in which every individual is definable withoutparameters, challenges this conclusion. In this article, I introduce a flexiblenew method for constructing pointwise definable models of arithmetic and settheory, showing furthermore that every countable model of Zermelo-Fraenkel ZFset 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 desiredindividual $a_n$ definable at each stage $n$, while preserving thesedefinitions through to the limit model, which can thus be arranged to bepointwise definable. A similar method works in set theory, and one can moreoverachieve $V=L$ in the extension or indeed any other suitable theory holding inan inner model of the original model, thereby fulfilling the resurrectionphenomenon. For example, every countable model of ZF with an inner model with ameasurable cardinal has an end extension to a pointwise-definable model of$\text{ZFC}+V=L[\mu]$.

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