Reddit reviews Understanding the Infinite

>I take it you're saying there are other axiomatic systems which also have value where things behave differently?

There are two ways of thinking about it. On the one hand yes, there are weaker axiomatic systems that recover much of our mathematics, the study of which is called reverse mathematics. The big text on this is Subsystems of Second Order Arithmetic. In reverse mathematics, we study which axioms are needed for individual mathematical results. As it turns out, usually very weak systems of arithmetic suffice, but if our systems are weak enough, what we end up with is revisionary mathematics, in which we lose some theories (for instance, without the Weak Konig's Lemma we lose that a continuous real function on any compact separable metric space is bounded). In some of these weak systems of arithmetic, the world of mathematics is finite (or, at least, it appears finite from stronger systems). That is true for instance in Robinson's Q, in which we can't even prove N != N + 1 for all N. Note however that this finitism is only apparent, so that if there is a fact of the matter regarding which system of arithmetic holds for mathematics, and that system is finite, we might still be able to do mathematics involving 'infinite' cardinals, but where such theories are satisfied by intuitively 'finite' models (as you can imagine, this gets philosophically tricky). Parsimonious considerations, along with the physical impossibility of manifested infinities in the real world, have led many to be classical finitists along these lines.

On the other hand, we can think of finitism as a meta-mathematical position, which may or may not be revisionary. A revisionary approach is Sazonov's feasible numbers, and a non-revisionary approach is given by Shaughan Lavine in Understanding the Infinite in which he recovers all our infinitary semantics, systems including large cardinal axioms, anything whatever in set theory, by appeal to the concept of indefinitely large sets as a substitute for infinity.