Axiom of infinity exceptions?

Question

It is my understanding that the axiom of infinity, or axioms that depend on the axiom of infinity, are the only axioms that are not representable in finite First-Order Predicate Logic. Is this correct? If so, how can the axiom of infinity be justified? Furthermore, it is my understanding that the axiom of infinity requires infinite induction, which requires the axiom of infinity. How is this circular definition resolved? Is there a way for the axiom of infinity to be defined so that it does not require infinite induction?

Answer 1

The axiom of infinity can be represented perfectly well in first-order logic, for example by asserting the existence of an inductive set (a nonempty set closed under the successor relation, like $\omega$). This does not require "infinite induction": it's just an ordinary statement in the language of set theory stating that something satisfying a certain condition exists. There is no circularity in the definition or the axiom.

The question of justifying the axiom is a different matter. One approach would be to say simply that lots of mathematics we want to do requires this axiom, so we assume it. Most mathematicians are happy enough with this pragmatic position, although it leaves unanswered the question of whether the axiom of infinity, like the other axioms of set theory, is just an assumption under which we carry out mathematical reasoning or whether we believe that there really are an infinity of sets.

Answer 2

While it is correct that generally it is impossible to write a sentence in first-order logic whose content is "there are infinitely many objects in the universe", we can still - under some circumstances - write a sentence which can prove that there are infinitely many objects.

For example, in a language which has a binary relation symbol $<$ we can write the following axioms (and take their conjunction, which is one sentence):

1. $\forall x(x\nless x)$ ($<$ is irreflexive).
2. $\forall x\forall y\forall z(x<y\land y<z\rightarrow x<z)$ ($<$ is transitive).
3. $\forall x\forall y(x<y\rightarrow\exists z(x<z\land z<y)$ ($<$ is dense).

From the conjunction of the three sentences we have one axiom whose implications are, amongst other things, that there are infinitely many elements in the universe. Formally, of course, we don't prove just one sentence, we prove a whole schema. We prove that for every $n\in\Bbb N$, these three prove that $\forall x_1,\ldots,\forall x_n\exists y(\bigwedge_{i=1}^n x_i\neq y)$. So no matter what $n$ elements we take, there is one more. Therefore there can't be finitely many elements in the universe.

Similarly, the axiom of infinity is a first-order axiom which states that there exists a set with certain properties. From the existence of that set we can prove that the universe contains a set which is not finite.

Also see Set Notation (Axiom of Infinity).