FOM: Large Cardinals Axioms

Harvey Friedman friedman at
Thu May 11 10:45:17 EDT 2000

This is a response to Avron 1:33PM 5/11/00.

Avron quotes me as follows:

>"Because of the thematic nature of these developments, and the interaction
>with nearly all areas of mathematics, large cardinal axioms will be begin
>to be accepted as new axioms for mathematics - with controversy. Use of
>them will still be noted, at least in passing, for quite some time, before
>full acceptance. This was the case with the axiom of choice. This will be
>accompanied by various improved intrinsic justifications of large cardinals
>along various lines."

I reiterate my belief in this statement.

>I believe that the comparison with AC here is inappropriate. The big
>is that AC has implicitly been used by mathematicians (at least in a weak
>form, like DC) long before it was explicitly formulated by Zermelo and Russel.

I agree that this is a difference between AC and large cardinals. However,
I was talking about what I anticipate to be the similarity.

>...In contrast, I dont believe that any "core" (or other)
>mathematician has ever implicitly used  large cardinal axioms in her/his
>work, and I dont believe one ever will...

But eventually they will be used so frequently and effectively in so many
concrete contexts, over and over again, that people will stop bothering to
explicitly mention them. However, as I said, this will have to be preceded
by a warming up period where it is expected to be explicitly mentioned.

>  Let me add here that personally I cannot be persuaded by Friedman's
>(and impressive achievements) because of my (old-fashioned?) criterion of
>accepting a statement as an AXIOM only if it is obviously, self-evidently

You are not talking about axioms, but rather "intrinsic axioms," which is
to be distinguished from "extrinsic axioms." I think that the difference is
one of terminology. Perhaps you would rather use terminology like "commonly
accepted principles."

>And I dont see what can make me suddenly realize that a certain large
>cardinal axiom is in fact self-evident (how could I have missed such a
>self-evident fact before??

I was not talking about self-evidence. This goes on throughout science -
that principles become universally accepted without being self evident. The
analogies with physical science are certainly not perfect, but some
analogies with physical science work well.

>Worse: I dont see what can possibly convince
>me that large cardinal statements are meaningful - let alone true...
>What I can at most accept, on empirical ground, is that many of them
>are most probably consistent, and so what can be proved using them
>most probably cannot be refuted. But for this what is actually used is
>the consistency of these axioms, not the axioms themselves.

So you may want to accept reflection principles on large cardinals. Of
course, from a practical (not foundational or philosophical) point of view,
this is in essence equivalent to accepting the large cardinals themselves.

> So my point is: arguments like those of Friedman can be very convincing, but
>only for scientists who are already more or less convinced.

No, because my main thesis is that we need to deal with large cardinals one
way or another in order to properly develop some important thematic
illuminating beautiful interesting informative valuable concrete
mathematics of very wide scope. And the inevitable way of dealing with
large cardinals for this purpose is to accept them - either literally, or
at least with reflection principles (which, practically, amounts to the
same thing).

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