Why accept the axiom of infinity?

If you believe in the set of natural numbers you already accept the axiom of infinity. For most mathematician existence of the set of natural numbers is an intuitively clear fact that doesn't need an argument so mathematicians are typically not bothered with the axiom. In addition lots of classical mathematics depends on such infinite concepts.

The question of accepting or rejecting such an axiom is mainly interesting for philosophers not mathematicians. One can reject the axiom of infinity (such people are often called finitists) but most mathematicians do not. They believe in the existence of the set of natural numbers and therefore see the axiom of infinity as a trivially true fact.

It is possible and interesting to have as our axiom system the usual ZF, but with the Axiom of Infinity replaced by its negation. The resulting theory, which one might call the theory of (hereditarily) finite sets, turns out to be bi-interpretable with first-order Peano arithmetic.

A finitist, who rejects the axiom of infinity, will be denied both the Dedekind and Cauchy constructions of real numbers. The problem? The reals are uncountable (Cantor showed this), and in a finitist's universe, the universe itself is countable.

$V_0=\varnothing$

$V_{n+1}= \mathscr P (V_n)$ where $\mathscr P$ denotes the powerset.

The universe if we accept the negation of the axiom of infinity is $V_\omega=\bigcup_{x\in\omega} V_x$, a countable union of at most countable sets.