Re: Some grey areas in foundational issues

From: george (greeneg_at_cs.unc.edu)
Date: 03/12/05 

Date: 12 Mar 2005 09:53:03 -0800

Aatu Koskensilta wrote:
>
> The compactness theorem is about standard proofs,
> not non-standard proofs. For non-standard proofs
> you get a non-standard compactness theorem:
> if there is a non-standard proof from an infinite set of
> premises P then there is a non-standard proof from a
> subset of P of possibly non-standard cardinality.

PA can prove a lot of important things about PRA
(such as that PRA has two models disagreeing about
the truth value of Con(PRA), via G1), but does
it follow from that that it can prove the existence of
"cardinality"? In the standard model of PRA, each pair
of different "things" has a different cardinality,
so you might as well identify the cardinality of any
thing WITH that thing. In the standard model, although there
exist subsets whose cardinality is infinite, that infinity
is not present in the domain of the model, so it is hard to
see how to found any notion of cardinality at all. In the
non-standard models, however, supernatural terms are available
and they could be identified with the cardinalities of infinite
sets. So how is that normally done? In ZFC, cardinals are
reduced to ordinals. Is there some intermediate notion LIKE
"ordinality" that could be replicated in PA-as-a-meta-theory,
when reasoning about PRA~Con(PRA)-as-an-object-theory?
Or is that intermediate step (via ordinals to cardinals)
skippable in the simpler theory?

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