[FOM] The defence of well-founded set theory

[Aatu Koskensilta]

On Oct 3, 2005, at 11:16 AM, Roger Bishop Jones wrote:

> On Thursday 29 September 2005  9:54 pm, Stanislav Barov wrote:
>
>> H. Wayl
>> influented intuitionistic criticue formulated his own version
>> of set theory which was precisly described and improved by S.
>> Faffermann. As sone in ZF was obvious lack for big sets Geodel
>> and Bernais make defenitions for notion of class which isn't
>> set and not contained in other class.
>
> Its not clear to me what cogent critique of the iterative
> conception these respond to.
> (indeed NBG might be, perhaps usually is, considered consistent
> with that conception)

NBG is consistent with the usual conception of set theoretic hierarchy 
in a rather trivial sense: one can consider the totality of classes to 
consist of properties or collections definable in the language of set 
theory. The intelligibility of the language of set theory itself seems 
to imply the acceptability of such a totality of classes. The only 
controversial thing in NBG not motivable in this fashion is the global 
axiom of choice for which you need to use forcing to prove 
conservativity. I'm not sure how commonly the global axiom of choice is 
actually considered to belong to NBG, but it is included at least in 
the original von Neumann formulation (or, rather, is an immediate 
consequence of the limitation of size principle).

NBG is of course only one first step in the sequence of "predicative" 
class extensions of ZFC arguably as acceptable as ZFC and the notion of 
set theoretic truth. These theories are all weaker than ZFC+"There is 
an inaccessible" (in the sense that ZFC+IA proves every set theoretic 
sentence provable in any of these extensions of ZFC).

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

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