I was reading on first and second order logic (The first has quantifiers over individuals of the domain and the second can have quantifiers over predicates as well - see this question: Differentiating First/Second order logic) and saw these two wikepdia pages:
https://en.wikipedia.org/wiki/Second-order_logic#Non-reducibility_to_first-order_logic-> With a section about how you can't convert the least upper bound property of the real numbers into FOL.
https://en.wikipedia.org/wiki/Peano_axioms#First-order_theory_of_arithmetic -> With a section on how to axiomative the induction axiom of Peano Arithmetic (a statement in SOL) into a FOL axiom schema.
So it seems that sometimes you can convert sentences in SOL into FOL and sometimes not. Is there a result that tells you when this conversion is possible?
My background is introductory courses to logics and set theory and a chapter on Recursive functions, from the book "Logic and Complexity" by Richard Lassaigne.
Thanks in advance