[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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