Are there any coherent, complete, mathematical systems that do not imply the existence of the infinite?

Question

Basically, are there any systems(complete with axioms) that do not imply the existence of an infinity? Is it possible to construct a mathematical system without infinity?

Answer 1

First-order theory of (lots of things there is a first-order theory of), if it's complete (I'm assuming you mean complete in the sense of mathematical logic) and doesn't have a predicate "is infinite". Real closed fields, for example.

Just to clarify: any model of the theory of real closed fields is infinite, and for any positive integer $n$ there is a theorem that says the field $F$ has at least $n$ distinct elements, but the statement "$F$ is infinite" is not part of the theory.

Answer 2

If you take the axioms of ZFC and replace the Axiom of Infinity with its negation, you get the hereditarily finite sets. This is a consistent set theory in which you can model each of 0, 1, 2, 3, 4, ... But there is no set containing all of them.

In fact it's precisely the Axiom of Infinity (in standard math) that allows us to form the completed set of natural numbers. Without a special axiom, we can't do it.

The reason most mathematicians accept the Axiom of Infinity is for convenience. It's hard to do math without infinite sets.

The question as to whether the Axiom of Infinity is true or false in some Platonic sense is a matter of philosophy. In fact the Axiom of Infinity is exactly what gives us an actual infinity in the sense of Aristotle.

In math we are free to take axioms for convenience. Since it's far easier (and way more fun) to do math with infinite sets, that's the choice that's made.