[FOM] RE: FOM Large Cardinals with Nonstandard Methods?
Matt Insall
montez at fidnet.com
Wed Jul 30 21:44:50 EDT 2003
[Friedman]
In particular, one would like to be able to derive or at least prove the
consistency of certain large cardinal axioms either thinking nonstandardly,
or thinking about standard interpretations nonstandard ideas.
[Insall]
This I cannot see at all. Nonstandard methods are are derived in
set theoretic terms in a theory like ZFC. In order to prove the
consistency of a large cardinal axiom using nonstandard methods,
the set theoretic axioms one uses to construct the nonstandard models
need to be at least as strong as the consistency strength of the large
cardinal axiom in question. There is little I can see that is gained
by saying ``thinking nonstandardly'', for in fact, a main contribution
of Robinsonian nonstandard analysis is that it encodes in ZFC (or a similar
collection of set theoretic axioms) the informal intuition behind
``thinking nonstandardly''. Another main feature of nonstandard methods
that has been mentioned several times in this discussion is the fact that
the theories developed in nonstandard mathematics are conservative
extensions
of the standard theories they extend. Thus ``thinking nonstandardly''
cannot result in a proof of any theorem about the original structures
that is not provable in the original theory. On the other hand, thinking
nonstandardly may provide us an intuitive reason for believing a particular
large cardinal axiom or its negation. I guess this might happen if a
particular
result of, say, analysis, is particularly clear using nonstandard methods
and, say
an inaccessible cardinal, but appears (or is provably) intractable without
an
inaccessible cardinal.
[Friedman]
I don't see how to do this, but I have made a step towards this in the
context of general nonstandard mathematics.
[Insall]
Scanning the rest of your post, I see what you mean by ``general nonstandard
mathematics''. It is interesting, but is not quite what I mean by
``nonstandard
methods''. I consider the elementary equivalence of the nonstandard
extensions
to be fundamental, and your approach to ``general nonstandard mathematics''
removes the requirement of elementary equivalence for the extensions.
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