Differences between set theory and second order logic?

Question

In first order logic only $x$ in $p(x)$ is quantified but in second order logic it is also possible to express quantified predicates. Set theory is defined in first order logic, as far as I understand, but have a nature of being of second order since both sets and their elements can be quantified.

Which are the characteristic differences between set theory and second order logic? Could secondary logic supersede set theory in mathematics?

How do you define second-order logic properly without having some notion of semantics and sets? — Asaf Karagila, Jul 19 '15 at 11:31
@AsafKaragila. I have no definition. I refer to https://en.wikipedia.org/wiki/Second-order_logic and only mean the possibility of quantifying predicates, as $\forall P\exists x: P(x)$. — Lehs, Jul 19 '15 at 16:40
You can see Stewart Shapiro, Foundations without Foundationalism : A Case for Second-Order Logic (1991), for a detailed "foundational project" based on second-order logic. See W.V.Quine, Philosophy of Logic (2nd ed, 1986), page 64-on, for a "classical" critique od SOL as "set theory in sheep's clothing". — Mauro ALLEGRANZA, Jul 20 '15 at 08:45
You may want to read http://plato.stanford.edu/entries/logic-higher-order/. — user21820, Aug 01 '15 at 08:30
@AsafKaragila You could define it by saying that if you replace $P(x)$ in $\forall P\exists x: P(x)$ with an arbitrary formula that may contain instances of $x$, that the original proposition implies the new proposition. — Paul, Feb 21 '18 at 00:09

Differences between set theory and second order logic?

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