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Stan wrote:
[Selected remarks]
> vfilipch ha scritto:
> > If "yes", does it mean that there are some restrictions on elements in
> > the Set Theory?
>
> Essentially yes. Set Theory is an AXIOMATIC system set-up to
> encode in a compact form what is called mathematical logic or
> Aristotelian logic (there are people trying to propose/develop other
> logic systems).
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.
> As such, given that the axioms are conflict-free,
Many people at least ESTIMATE that set theory is consistent.
> Set Theory can be neither proved nor disproved.
Sentences are proven from axioms. If 'prove a theory' had a meaning, it
would be that of proving all the sentences in the theory. But then set
theory is proven, as each theorem, is by defintion, proven by the
axioms.
> It defines sets and set elements indirectly by means of the
> axioms and, in principle, does not care whether out there in the
> real world exists anything compatible with them (in practice,
> of course, the real-world rules of logic were used as a MODEL
> to construct it).
I think set theorists do hope that the defined objects of set theory
correspond in certain ways with what are considered by many people to
be mathematical objects, such as numbers.
> It requires very little of the set elements - essentially just that
> they
> can be uniquely referenced (labelled), distinguished from each other,
> and, if need be, associated into n-tuples which then can be
> treated as new elements.
That objects can be associated into tuples is derived in Z set
theories, but (I think) it is axiomatic in certain other set theories.
MoeBlee
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