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I've been trying to get a handle on how higher order logic interacts with set theory. It's been stated convincingly that higher order logic with full semantics is set theory in sheep's clothing. For instance, an identity that is used for higher order Skolemization is equivalent to the axiom of choice.

If higher order logic with full semantics is just set theory, which set theory is it? NBG, new foundations, ZFC? Can we just pick any first order set of axioms for set theory and 'lift' them to higher order logic by interpreting the membership predicate? Where do the full semantics come from?

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    $\begingroup$ Where has it been stated convincingly that higher order logic is set theory in sheep's clothing? Quine, certainly (but how convincingly), but who else? $\endgroup$ – Thomas Benjamin Dec 29 '14 at 12:39
  • $\begingroup$ It looks fairly obvious when you can replace membership relations with monadic higher order predicates that the shape is the same, and I'm not certain but I would find it entirely unsurprising if its been proven that there are isomorphic models between higher order logic and set theory. It looks like set theory, type theory, and higher order logic all are the same thing with different clothes (syntax.) $\endgroup$ – dezakin Dec 30 '14 at 20:18
  • $\begingroup$ @ThomasBenjamin I think the fact that e.g. there is a second-order sentence which is a validity iff CH is true, is a pretty convincing argument. $\endgroup$ – Noah Schweber May 6 '16 at 19:04

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