Re: Cantor's definition of set

On Fri, 26 Oct 2007 17:54:18 +0000 (UTC), Chris Menzel
<cmenzel@xxxxxxxxxxxxxxxxxxxx> wrote:


Notably, extensionality in this language is expressed
as:

forall r forall s forall z ((z in r <-> z in s) -> r=s) .

Sure? I'd rather write

forall r forall s (forall z (z in r <-> z in s) -> r=s) .

See:
http://plato.stanford.edu/entries/set-theory/ZF.html


A many-sorted first-order logic can always be "reduced" to a
single-sorted logic by introducing a distinct predicate for each sort --
in this case, say, "Set" and "Urelement" -- and translating each
many-sorted sentence into a sentence with appropriately relativized
quantifiers. Your axiom above in particular would be the single-sorted
counterpart of extensionality.

Yes. As has been shown be Arnold Schmidt (1951).

Arnold Schmidt, Die Zulässigkeit der Behandlung mehrsortiger Theorien
mittels der üblichen einsortigen Prädikatenlogik. (1951)


F.

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