Equality is the intersection of all equivalence relations.
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Replying to @ModelOfTheory
Not just that, but even more simply, equality is the intersection of all reflexive relations.
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Replying to @RadishHarmers @ModelOfTheory
(This is essentially the adjoint characterization of equality, in the approach where all logical connectives are understood as given by adjoint functors)
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Replying to @RadishHarmers
I spoke of equivalence relations rather than reflexive relations because, although reflexive relation is the most general that makes the claim true, the equivalence relation axioms form a complete theory if = is suppressed from the language, and this is the theory satisfied by =.
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For anyone following along, RadishHarmers pointed out to me that the theory of an equivalence relation is not complete, even when = is suppressed from the language. It is, however, the theory of = for a set of unknown cardinality.
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