  Throughout the course of mathematical history, generalizations of previously
understood concepts and structures have led to the fruitful development of the
hierarchy of number systems, non-euclidean geometry, and many other epochal
phases in mathematical progress. In the study of formalized theories of
arithmetic, it is only natural to consider the extension from the standard
model of Peano arithmetic, $\langle \mathbb{N},+,\times,\leq,0,1 \rangle$, to
non-standard models of arithmetic. The existence of non-standard models of
Peano arithmetic provided motivation in the early $20^{th}$ century for a
variety of questions in model theory regarding the classification of models up
to isomorphism and the properties that non-standard models of Peano arithmetic
have. This paper presents these questions and the necessary results to prove
Tennenbaum's Theorem, which draws an explicit line between the properties of
standard and non-standard models; namely, that no countable non-standard model
of Peano arithmetic is recursive. These model-theoretic results have
contributed to the foundational framework within which research programs
developed by Skolem, Rosser, Tarski, Mostowski and others have flourished.
While such foundational topics were crucial to active fields of research during
the middle of the $20^{th}$ century, numerous open questions about models of
arithmetic, and model theory in general, still remain pertinent to the realm of
$21^{st}$ century mathematical discourse.
