I'm reading the Stanford Encyclopedia of Philosophy entry on Second-order and Higher-order Logic. In it, I read the following:
[W]e can express the Peano induction postulate by a second-order sentence:
∀X[X0 & ∀y(Xy → XSy) → ∀y Xy]
This sentence expresses the idea that X is true of all natural numbers, if it is true of 0 and its truth at some number y guarantees its truth at the successor of y, no matter what set of numbers X might be true of.
Now, I don't quite see why this sentence has to be second-order to express the Peano induction. That is, what would be missing if it was written in first-order logic, that is, without the "∀X" part?
X0 & ∀y(Xy → XSy) → ∀y Xy
I'd read this in the following way: IF (X is true for zero) and (for every y if X is true for y then it is also true for the successor of y) THEN X is true for all y's. What seems to be missing from my natural language interpretation of the modified sentence compared to the quoted interpretation of the original sentence above is the "no matter what set of numbers X might be true of", but I don't really see what this adds. Is it implied in my first-order sentence that this does not hold for certain sets of X? If so, I really can't see it.