Talk:General set theory

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[edit] existential axioms

Maybe it's just me, but i think the axiom of separation assures the existence of the empty set *if a set already exists*. None of these axioms assures the existence of any set, since their first quantifier is universal, so they suppose a set already exists. It must be just me anyway, because i can't imagine that GST would be mentinned if it really had the flaw of reasoning on entities while being unable to prove their existence... i'm still puzzled, though. Spiritofhere 09:54, 10 August 2007 (UTC)

I am not familiar with Boolos' work, but it is common to take \exists x ( x = x) as an axiom of logic. Perhaps that is the case here. — Carl (CBM · talk) 13:00, 10 August 2007 (UTC)
Separation assures the existence of an empty set in GST in the same way that existence is assured in ZF. Just let φ(x) be always false. Starting from the empty set, Adjunction assures the existence, by elementary recursion, of the sets needed for von Neumann's ordinals and Peano arithmetic.
Carl, what you call an axiom of logic strikes me as an assertion that the domain is nonempty. No problem in this corner, as the logic of empty domains ("free" logic) has always struck me as vacuous, pedantic, and uninteresting. Your assumption is part of the bare minimum needed to get mathematics up and running.202.36.179.65 06:53, 15 September 2007 (UTC)