My logic course points out that most statements of mathematics can be given by first order logic, but that the completeness axiom for R needs to be quantified over sets and so uses second-order logic.
I'm wondering what this implies for the completeness axiom - is it a different kind of mathematical statement to smaller, simpler ones? Does this effect the validity or perceived epistemic value of maths in some way that it wouldn't if this were phraseable in first-order logic?
Wasn't sure whether to put this is in philosophy or maths so sorry if i chose the wrong one!
First of all, note that we are talking about first- and second-order logic over the signature of $(R, +, -, \times, 0, 1, <)$ of real ordered fields. In this language, the ordered field $\mathbb R$ can be given as the unique structure satisfying the first-order axioms of an ordered field (things like $a + b = b + a$ or $a \neq 0 \to \exists b(ab = 1)$), together with one additional second-order axiom: the axiom of completeness.
When we want to formulate mathematics as a whole, not just the real numbers, in a logical language, this is typically done ZFC (or a similar set theory). This is a first-order theory whose domain is that of all sets. In this language, stating that $\mathbb R$ is complete is an ordinary first-order sentence: quantification over sets is all that first order ZFC sentences do.
It is in the basic sense that the completeness axiom for $\mathbb R$ cannot be stated in the first order theory of real ordered fields. One way to see this is that, by Löwenheim-Skolem, all first-order theories with infinite models must have infinite models of all cardinalities, while the only model up to isomorphism of the ordered field axioms + the completeness axiom is the real numbers.
One way of phrasing this is: first-order axioms are too weak to pin down a single infinite model, while second-order axioms are strong enough to do so.
Because of the "in some way", this is a very wide question, but as an attempt at an answer: the vast majority of mathematicians accept some form of first-order set theory as a valid foundation of mathematics. Within that set theory, the completeness axiom is just a first-order sentence, and thus perfectly valid and epistemically sound.