I'm currently going through Enderton's Elements of Set Theory. I have two questions, but they need some context. Here are the subset axioms introduced in the book:
For each formula _ not containing B, the following is an axiom: $\forall t_1 \dots \forall t_k \forall c \exists B \forall x (x \in B > \iff x\in c $ and _ )
In English, the axiom asserts (for any $t_1, \dots, t_k$ and $c$) the existence of a set $B$ whose members are exactly those sets $x$ in $c$ such that _.
My questions:
Can these subset axioms be used with an infinite number of $t$'s? I'm confused because Enderton uses these axioms to justify the intersection of an infinite collection of sets. If your collection is $A = \{b_0, b_1, b_2, \dots\}$, and you want to create an intersection of the sets in $A$, what application of the subset axiom are you using? What is $k?$
How do we know infinite sets are legal things? I'm assuming this sort of thing has to be justified or stated as an axiom, but it isn't. Enderton just writes in an earlier part of the book "But suppose we want the union of infinitely many sets; suppose we have an infinite collection of sets $A$".
Here's a small hint (maybe). The $t_i$'s are just variables appearing in the formula which defines the subset $B$ of $c$. It wouldn't make sense to have an infinite number of $t_i$'s because formulas are finite in length. In particular, a formula expressing that $x$ is an element of every set in a (possibly infinite) family, can be written down with finitely many symbols.
And yes, the existence of at least one infinite set is generally assumed as an axiom (e.g. ZFC's axiom of infinity). I'm not familiar with the book you're using so I can't explain why it doesn't appear there.