[FOM] Question about Set Theory as a formal basis for mathematics

[Aatu Koskensilta]

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