FOM: Factoring axiom vs Large cardinal axioms

[Harvey Friedman]

Reply to Marker 6/15/00 11:16AM and the correction of 6/12/00 2:06PM.

>I found the "Does Mathematics Need New Axioms" panel discussion in Urbana
>was quite interesting and thought that all of the panelists did a great
>job explaining their positions in the limited time available.
>
>While the discussion focused on set theoretic (in particular large
>cardinal) axioms, elsewhere at the meeting Avi Widgerson discussed
>the consequences of the "multiplication is hard axiom". While this
>is an assertion of a very different flavor, it has been
>enthusiasticly embraced and all of us rely on it to an extent that it
>would now have dire consequences if it were false.
>
>I would be interested in hearing the panel's comments on this.

>In my earlier message "multiplication is hard" should have
>been "factoring is hard"
>
>Sorry for the confusion.

This is a very good issue - i.e., a detailed comparison between the
factoring axiom and large cardinal axioms. Quite a lot can probably be
learned by going into this comparison in detail.

At first blush, most logicians would say that they are completely
different, entirely non analogous. But I think that Marker is implicitly
emphasizing possible similarities.

I will divide my discussion of this into similarities and differences. I
find that the dissimilarities are much more substantial and deeper than the
similarities.

SIMILARITIES BETWEEN THE FACTORING AXIOM AND LARGE CARDINAL AXIOMS.

Experts in the two relevant fields both almost universally accept these
axioms, at least for the smaller of the large cardinal axioms. But for the
larger of the large cardinal axioms, acceptance is much less widespread.
This imeediately leads to the question of whether there are strong forms of
the factoring axiom that similarly lose strong adherents. Or, more
interesting still, perhaps a hierarchy of stronger and stronger forms of
the factoring axiom that lose more and more strong adherents?

In both cases, a principal component of the confidence the experts have
comes from experience with using the axioms, drawing interesting
consequences that are deemed plausible and desirable, and also in the
course of this, utterly failing to refute the axioms.

There is also a similarity from the point of view of the informed outsider.
The informed outsider readily grants that for the purposes of both fields,
these axioms are entirely appropriate working assumptions. The informed
outsider generally does not share the confidence that the insiders have
that these working assumptions will not have to be eventually discarded.

DISSIMILARITIES BETWEEN THE FACTORING AXIOMS AND LARGE CARDINAL AXIOMS.

1. Large cardinals are presented by experts as an intellectually natural
and essentially unique extension of a process that is virtually universally
accepted by informed outsiders. Namely the virtually universally accepted
process of setting up the standard axioms for mathematics (ZFC).

2. Regardless of how one feels about 1, large cardinal axioms form a
compelling, robust looking, *hierarchy*, where one becomes totally familiar
with the process of picking a place where one is dubious about acceptance -
even dubious about formal consistency. For some people this point may be
way below ZFC, and for others, way above ZFC.

3. In light of 1 and 2 there are various formal results that state that you
cannot jump from one place to a higher place in this hierarchy.

4. These features do not seem to be present at all in the factoring axiom
situation.

5. It would be quite interesting to see whether the factoring axiom
situation could be made more analogous to 1-4 above. I.e., what a hierarchy
of statements would look like, and what kind of impossibility theorems
about jumping from one place to another would look like.