Are there infinite sets of axioms? I'm reading Behnke's Fundamentals of mathematics:

If the number of axioms is finite, we can reduce the concept of a consequence to that of a tautology.

I got curious on this: Are there infinite sets of axioms? The only thing I could think about is the possible existence of unknown axioms and perhaps some belief that this number of axioms is infinite.

An example of an infinite set of axioms used for practical use is the foundation of non-standard analysis.
Create a first-order language whose symbols include:


*

*Every element of $\mathbb{R}$

*Every element of $\mathcal{P}(\mathbb{R})$

*Every element of $\mathcal{P}(\mathcal{P}(\mathbb{R}))$

*:


Now select a set of axioms to be the set of all true statements in this language.
This language and set of axioms is enough to do lots of things; e.g. any theorem you've learned in calculus can be stated in this language and is automatically a theorem.
By general properties of first-order logic, this set of axioms is consistent and thus there exists a non-standard model! By construction, every theorem you know about 'standard' things is automatically true for non-standard things as well! In this formulation, the transfer principle from non-standard analysis -- every 'standard' object has a nonstandard analog, and every statement of 'standard' analysis is true if and only if its translation to the non-standard model is true -- becomes an utter triviality.
While one can surely attempt to select a smaller set of axioms, there is no actual reason to; for the specific application we have to apply logic in this instance, it is by far easier and more convenient to take all of them.
