FOM: Large cardinals as integers?

Harvey Friedman friedman at
Fri Feb 18 10:19:29 EST 2000

Reply to Kennedy Fri, 18 Feb 2000 09:46.

>The Downward Lowenheim-Skolem Theorem gives a way of viewing, say,
>a measurable cardinal, as a natural number. Does anyone think that this
>fact is relevant to the recent debate on fom about new

Not me..

>Also, does Harvey think that this fact has any confirmatory
>value with regard to his program involving large cardinals and finite
>combinatorial statements?

No. The only relevance I see is that when I show that various statements in
discrete mathematics are independent of various very strong formal systems
(although the ones that are really natural are all at the moment proved
well below measurable cardinals, which is a medium large cardinal), I start
with the combinatorial statement and then construct a model of set theory
with some small large cardinals. The model must have nonstandard integers,
and I construct a countable model. The small large cardinals appear as
points in a countable structure.

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