Analytic method for number theory-do we have to assert second-order logic? I am an undergraduate. I am just starting to study logic and analytic number theory at the same time, so please forgive me if I made an elementary misunderstanding.
A lot of theorem in number theory can be stated completely in term of first order logic added with Peano axiom. But they were proven using analysis, and for many of them we do not know of any elementary method to do so, which means that second order logic needed to be used.
So I have 3 questions:
-Theorem in number theory need first order logic to be stated, so axiom of first order logic are assumed. But since their proof use second order logic, does that means that these theorems are, essentially, proved by asserting new axioms (ie. axioms of second order logic)? Would this pose any problems?
-Is there any theorem in number theory that is proved, and is also proved that all possible proof of that theorem must use more than first order logic?
-If we do not accept the axioms of second order logic, is it possible to construct a system of number satisfying Peano arithmetic such that: there is no possible way of using analysis? some of those theorem proved using analysis are false?
 A: *

*mathematical proofs are generally/mostly thought of "living" in zfc (zermlo fraenkel & choice) set theory. If principles are used that go beyond that theory, then typically this needs to be made explicit. So, proving an "arithmetical" formula with means of zfc is a valid mathematical argument.

*the field in which people try to find out which portion of analysis(i.e. second order number theory) are necessary to prove some statements in arithmetic is called reverse mathematics. See also wikipedia

*to prove that a given part of second order arithmetic is necessary to prove a theorem at hand, one uses the theorem to prove the axioms of the given system - thus the name "reverse" mathematics.

A: It often happens that number-theoretic statements or proofs that use complex analysis and therefore appear to be second-order over arithmetic can be reformulated in first-order arithmetic.  The techniques used for this include approximating real numbers with rational ones and approximating integrals with Riemann sums.  For example, the Riemann hypothesis can be reformulated in the language of first-order arithmetic.  (I confess that I'm not an expert on such reformulations; in particular, I've never worked through the details of converting the  Riemann hypothesis to first-order.)
