[FOM] Question about Set Theory as a formal basis for mathematics
Aatu Koskensilta
aatu.koskensilta at xortec.fi
Sat Feb 25 20:05:32 EST 2006
On Feb 25, 2006, at 4:23 PM, Andrea Proli wrote:
> So, the semantics of ZF is given in terms of what ZF itself defines?
> Or am
> I simply confused?
Yes, and you're not alone. ZF(C) does not define anything, it is just a
bunch of first order axioms. This bunch of axioms is foundationally
interesting since it codifies - as well as the first order language of
set theory allows - certain set theoretic principles which are
sufficient to prove most mathematical theorems, the principles having
been discovered partly by analyzing mathematical practice (of early set
theorists and mathematicians). The semantics of the first order
language of set theory in which these axioms are formulated in is given
by saying that the quantifiers range over the so called cumulative
hierarchy of sets obtained by starting with the empty set and iterating
the powerset operation "as long as possible". This hierarchy, and the
concepts necessary to understand its description, are not in defined by
ZF(C) but must be understood the same way one understands what the
natural numbers are, what a tree is and what this or that mathematical
concept means, whatever that way is.
In mathematical logic, one sometimes studies sets or structures (in the
cumulative hierarchy) which happen to be models of the ZFC bunch of
first order axioms, a bit like one studies structures satisfying the
group axioms in group theory. However, these structures have very
little to do with the meaning of set theoretical assertions and don't
enter in any way into ordinary mathematics. In particular, these
structures are wholly irrelevant when it comes to understanding the
axioms of set theory, since these axioms simply don't concern models of
ZF(C).
Aatu Koskensilta (aatu.koskensilta at xortec.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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