> In fact I can use induction to show that actual infinity is nonsense:
Yes. But, you must do it for each and every mathematical statement individually.
What follows is the explanation for the first-order axiomatization of Peano arithmetic from wikipedia.
If you even had any concept of the ideas you try to purport, you might have some conception of how mathematics addresses questions, how others have investigated these matters, and how ridiculous statements such as the one above appear.
The first-order induction schema includes every instance of the first-order induction axiom, that is, it includes the induction axiom for every formula phi.
This schema avoids quantification over sets of natural numbers, which is impossible in first-order logic. For instance, it is not possible in first-order logic to say that any set of natural numbers containing 0 and closed under successor is the entire set of natural numbers. What can be expressed is that any definable set of natural numbers has this property. Because it is not possible to quantify over definable subsets explicitly with a single axiom, the induction schema includes one instance of the induction axiom for every definition of a subset of the naturals.