Peanos axioms provide a way to describe and construct a set of natural numbers without ever mentioning numbers or their properties (+, -, next, etc...). Can the same, or something similar, be done with the idea of equality?
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3Do you mean "Peano"? If so (and if not, what do you mean?), in what sense don't those axioms mention numbers and their properties? (That said you may be interested in the discussion here.) – Noah Schweber Aug 14 '21 at 07:33
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1@user953376 I've never studied logic at that level. Out of curiosity, what's the first order phrasing for equivalence? – Alan Aug 14 '21 at 08:08
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2Yes. All mathematics can be done in a suitable foundational system, such as the Fitch-style system at "Predicate logic: How do you self-check the logical structure of your own arguments?". As with all Fitch-style systems, it has the rules =intro and =elim that govern equality. In fact, Peano Arithmetic relies on those as well; it is wrong to think that Peano managed to describe natural numbers without equality. – user21820 Aug 14 '21 at 14:02