[FOM] The defence of well-founded set theory

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Thu Oct 6 16:46:24 EDT 2005

Quoting Roger Bishop Jones <rbj01 at rbjones.com>:

> Of course, all this changes if we compromise by
> interpreting set theory in one or more V(alpha)
> rather than in V as a whole.
> Then NBG (or even set theory in w-order logic)
> becomes just another language for talking about
> (interpretable in) part of V.

What seems to me not very natural in ZFC is the restriction 
of "quantification" only in postulating the existence of 
the set {x in a | F(x)}. That is, only the "quantification" 
over x is bounded by a. Why not all quantifiers in F and 
even in the whole theory and its logic are bounded? 
Let me call such style of a set theory where "everything" 
is bounded as bounded set theory (BST), like Bounded 
Arithmetic (BA). This idea was also used in Kripke-Platek 
set theory (if considered with restricted foundation or 
regularity axiom schema). Of course, the construct 
{x in a | F(x)} vs. {x | F(x)}, as a mere remedy against 
Russel paradox, does really help, but "ideologically" this 
seems only a "half-step". The idea seems was that quantifying 
over the "whole" universe of sets (what does it mean "whole"?) 
is something wrong, "illegal". Thus, let us go directly to a 
kind of BST based on bounded quantification rather than to ZFC. 
Then, we could additionally postulate the existence of a 
transitive set (like V(alpha) above) which could be considered 
as a universe for ZFC and over which we can now quantify quite 
“legally”. In fact, the universe for such a BST over which 
we do not intend to quantify (at least in the way how the 
full unbounded first-order logic allows) looks rather as 
something "cumulatively growing", "potential" and never to 
be completed -- I think this is a quite consistent view. 
Unbounded quantification can lead to unfortunate temptation 
to speculate about completed universe what is definitely 
something wrong. Thus, the style of BST seems to me more 
natural way of presentation of set theoretic ideas which 
loses nothing from the strength of ZFC or of any other its 
stronger extensions. 

In fact, I and my colleagues actually considered some versions 
of BST mainly from computer science perspective, but not only. 

Vladimir Sazonov

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