[FOM] Proli's Question about Set Theory

[Aatu Koskensilta]

On Feb 27, 2006, at 4:57 PM, Andrea Proli wrote:
> Two questions:
> 1) Why does the non-existence of a set of all sets create troubles with
> the fact that the domain (I suppose by "domain" you mean the universe 
> of
> discourse) of a model is a set?

In the intended interpretation of the language of set theory the 
quantifiers range over all sets so the universe of the intended 
interpretation is the collection of all sets. However, in model theory 
one usually requires the universe of a model to be a set, and there is, 
provably in ZF(C), no set of all sets (and there are no structures 
containing non-set collections, or indeed such collections at all, in 
ZF(C)). Thus the intended interpretation of ZF(C) is not a model. ZF(C) 
does have natural models which are of form <V_kappa, {<x,y> | x in y, 
x,y in V_kappa}> where V_kappa is the kappath level of the cumulative 
hierarchy with kappa strongly inaccessible. The existence of these 
models is not provable in ZF(C), and they do not define the 
interpretation of the language of set theory.

> 2) What do you say that ZF implies that there is "no set of ordered 
> pairs
> that can serve as the interpretation of the membership predicate"?

It just means that the collection E={<x,y> | x in y} is not a set. E is 
the intended interpretation of the epsilon predicate symbol in the 
langauge of set theory. If E were a set, the collection of all sets 
would also be set, which it, provably in ZF(C), is not.

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus