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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.
2) In this Country "Aristotelian logic" is a plain language synonym
for "mathematical logic". Just a convention, nothing much to do
with the guy.
> > 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.
As for the rest, I was just trying to sort out what really bothered
the man who started this thread. It seems that he is convinced
that if he can think up an "object" which does not fit Set Theory,
the latter will fall. Somewhat along the line: since we have found
experimentally that the sum of angles in real-world triangles is
not pi, Euclidean geometry is not true.
The answer is that Euclidean gemotry is still live and healthy
but it is not the one *applicable* to the real world.
Maybe that one day we will find something similar about
mathematical logic (we do not know where logic came from,
anyway - it just works). Only it will not happen the way
vfilipch imagines.
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