 With the negation of the Axiom of Infinity, we get the theory of the \(\omega\)-hyperuniverse, which is equiconsistent with second-order arithmetic, and so actually has a fair amount of mathematical strength. In this theory, the class of natural numbers (considered as finite ordinals) is not closed and acquires an extra element “at infinity” (which happens to be the closure of the class of natural numbers itself). Individual real numbers can be coded (using the usual Dedekind construction, actually) but the theory of sets of real numbers will begin to look quite different.