TITLE: Why are higher-order logics less well-behaved?
QUESTION [9 upvotes]: I've read about about higher-order logics (i.e. those that build on first-order predicate logic) but am not too clear on their applications. While they are capable of expressing a greater range of proofs (though never all, by Godel's Incompleteness theorem), they are often said to be less "well-behaved".
Mathematicians generally seem to stay clear of such logics when possible, yet they are certainly necessary for prooving some more complicated concepts/theorems, as I understand. (For example, it seems the reals can only be constructed using at least $2^{\text{nd}}$ order logic.) Why is this, what makes them less well-behaved or less useful with respect to logic/proof theory/other fields?

REPLY [2 votes]: A further sense in which Higher Order Logics with standard or saturated semantics (HOL, hereafter) are less well-behaved than First Order Logic (FOL, hereafter) is a direct consequences of the failure of Completeness (and thus, as explained in other answers, of Compactness). The set of logical truths and the set of correct claims of semantic consequence for these logics are not recursively enumerable.
FOL is Complete, yet not Decidable. So, we determine of an arbitrary sentences and sets of sentences of the language of FOL if those sentences are logical truths or if a set has a given sentence as a consequence. But, since FOL is Complete and proofs are finitely long, we can (in the mathematicians sense of "can") enumerate the proofs and inspect one by one, checking what sentence the proof shows as a theorem or what sentence the proof derives from what set of assumptions. This gets us a recursive enumeration of the truths and the sentence/set pairs that stand in the consequence relation. (This does not contradict the failure of decidability as we cannot conclude that since we've yet to come across a proof in out enumeration, there isn't one if only we kept looking.)
Since HOL's are not Complete, this means of showing them recursively enumerable is not available. Indeed, there can be no means; were there such a means, it could be exploited to induce a Complete proof system, and there cannot be such a Complete proof system as the HOL's are not Compact.