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Stan wrote:
> MoeBlee ha scritto:
>
> > One of the things set theory does is to provide a meta-theory for
> > mathematical logic, but, it seems to me, that the primary intent is to
> > provide a theory in which to provide an axiomatization for ordinary
> > mathematics. As I understand, Cantor's initial motivation was to meet
> > certain problems he encountered in the study of certain series. And, as
> > I understand, Zermelo's initial motivation was to provide an
> > axiomatization (at that time, and informal one) from which to prove the
> > well ordering theorem. But, it seems that more generally, the role of
> > set theory is to axiomatize ordinary mathematics (and it is also a
> > theory in which to study mathematical logic). Also, I wouldn't describe
> > the logic so much as Aristotelian as, at least in its bare-bone
> > essentials (the first order predicate calculus, though Frege's own
> > system is a type theory) as well as many other kinds of logic.
>
> 1) I was under the impression that after Russell and Goedel, the
> mighty drive towards axiomatization of the whole Math dwindled
> to a barely noticeable brook.
After the incompleteness theorem was known, of course we wouldn't
expect to have a complete, consistent, recursively axiomatized theory
that includes arithmetic. But that didn't cause an ebb in the study of
formal systems (I don't think), but rather gave rise to even more
study, especially as recursion theory and model theory were developed
after the discovery of the incompleteness theorem. Anyway, set theory
serves the purpose of axiomatizing ordinary mathematics. It's an
incomplete theory (if consistent), but still serves as an
axiomatization.
> 2) In this Country "Aristotelian logic" is a plain language synonym
> for "mathematical logic". Just a convention, nothing much to do
> with the guy.
Okay, then that's your useage. But you'll find that usually, people
take them not to be synonymous, but rather that Aristotelian logic (in
the plain sense of syllogisms, moods, etc.) is but a proper part of
mathematical logic as Aristotelian logic (in that plain sense) can be
formalized as part of the monadic predicate calculus, which is a proper
part of the predicate calculus..
> > > As such, given that the axioms are conflict-free,
> > Many people at least ESTIMATE that set theory is consistent.
> I guess we agree here. But if your "estimate" implies "esteem",
> than I do not have much of it. After all, incosistent theories would
> not be worth talking about. Also, unfortunately, we have several
> internally consistent sets of axioms for presumably the same
> thing so the situation is not really admirable.
I don't follow you. What consistent theories do you have in mind for
"presumably the same thing" as set theory? What do you mean "for the
same thing"?
MoeBlee
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