I am looking for papers and books that try to reduce set theory to logic. I know there have been attempts to reduce arithmetic to logic, but I am looking for texts that, by logic alone, prove some of the basic axioms of set theory, like the existence of the empty set, the existence of the unordered pair of two sets, the existence of the union of a set of sets, the existence of the powerset of a set, etc. Has there even been such an attempt to ground set theory in logic, perhaps not first-order logic, but second-order or higher-order logic?
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$\begingroup$ Apart from PM specifically, I'd suggest looking more generally at logicism in the philosophy of mathematics. $\endgroup$– blargonerJan 31 at 20:10
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2$\begingroup$ How do you distinguish between "reduce set theory to logic" and "define logic so broadly that it includes set theory"? $\endgroup$– user14111Jan 31 at 21:52
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2 Answers
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I may well be misinterpreting your question, but it sounds like you're asking if the axioms of set theory can be reduced to pure logic ... to which the answer is no: these axioms are not logical truths. There is no way to derive any of these axioms from purely logical axioms.
We use axioms of set theory in order to formalize and 'make hard' certain intuitive conceptions we have about sets. You could say the axioms describe a world of sets. And yes, we can use the language of logic to do this. But note, this language includes symbols that are not purely logical symbols, such as $\in$. And while the axioms including those symbols describe 'truths' about set theory as we see them, as far as pure logic is concerned there could be a completely different interpretation of the $\in$ predicate that doesn't work at all like we imagine $\in$ to work, and in which the axioms of set theory end up being false.
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The magic words are "Principia Mathematica" by Whitehead and Russell (as opposed to the one by I. Newton)
