[FOM] Large Cardinals and Extensions

[Harvey Friedman]

Reply to Insall Large Cardinals with Nonstandard Methods?

On 7/30/03 9:44 PM, "Matt Insall" <montez at fidnet.com> wrote:

> [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.
> 
>
In my posting, I discussed the following statement.

PROPOSITION. Every sufficiently large algebra has a proper extension that is
"similar".

The usual relation here is "elementary extension". This is proved by
compactness, and the break point is omega.
 
I showed that a relation so simple that does not even involve elementary
extensions, not even first order elementary equivalence, is sufficient to
derive a measurable cardinal. This is: having the same finitely generated
subalgebras. This is the formulation that I personally find most satisfying,
surprising, and novel, as it is purely "algebraic" with no use of languages.

I also remarked that if you use elementary equivalence, or elementary
extensions, for languages just a tiny bit stronger than first order logic,
then you also derive large cardinals. E.g., if you use elementary
equivalence for weak second order logic, or elementary extensions for weak
second order logic, then you generate measurable cardinals.

The break points in each case of these cases are the first measurable
cardinal, if it exists.

Also, in the above, you get equivalence with the existence of a measurable
cardinal.

If you use second order logic, then using elementary equivalence or
elementary extensions, you generate even higher (well known) large
cardinals. You also get break points and equivalence.

See the references I gave in that posting,

H. Friedman, Restrictions and Extensions,
http://www.mathpreprints.com/math/Preprint/show/index.htt

H. Friedman, Working with Nonstandard Models,
http://www.mathpreprints.com/math/Preprint/show/index.htt

The second reference has a complete proof of the finitely generated
subalgebra result, but does not mention even the weak second order logic
result. 

I have expanded the second manuscript to incorporate the WSOL situation. The
preprint may disappear from the preprint server for a couple of days, just
before it is replaced with the new version.

Of course, in each case, I am talking about extensions that do NOT allow for
nonstandard integers, assuming that the ground structure has the integers
with a successor function. So one should not call this kind of thing
"nonstandard analysis" or "nonstandard arithmetic", but rather "nonstandard
mathematics" or "nonstandard set theory".

Harvey Friedman