FOM: determinate truth values, coherent pragmatism

Harvey Friedman friedman at
Tue Sep 5 15:48:30 EDT 2000

>At 12:02 PM 9/5/00 -0400, in a very interesting posting, Harvey Friedman
>>I'm saying that the general mathematical community may be compelled to
>>accept some new axioms that are not self evident. But this is going to
>>happen only through what I call coherent pragmatism. Issues of truth will
>>not enter the picture, as far as they are concerned. Only the issue of
>>consistency will be of concern. The analogy with the situation in the
>>sciences breaks down right here.
>Since consistent axioms can have false arithmetic consequences, consistency
>is not enough.

I'm saying that since there is no method for spotting false arithmetic
consequences, consistency is all that you have.

This is in sharp constrast with the situation in science/experiments.

Whereas one can argue quite sensibly that certain arithmetic statements are
false that are in fact consistent with ZFC, the only cases I am aware of
are those that are refutable using large cardinals. So, let me put this in
a bit more focused way:

There is no method for spotting false arithmetic consequences of large
cardinals other than refutations from large cardinals, even if there were
false arithmetic consequences of large cardinals.

And since all axiom candidates on the table are known to have compatible
arithmetic consequences (one included in the other), and all useful axiom
candidates on the horizon have compatible arithmetic consequences, the
issue of false arithmetic consequences will in fact not arise in the choice
of one axiom candidate or another.

You can argue that the overriding question of whether the arithmetic
consequences of large cardinal axioms are true will remain even if
mathematicians are convinced that they are consistent. Well, it may remain
in a sense, but nothing is going to be done about it. There will be no way
to act on it.

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