[FOM] The Gold Standard

[Harvey Friedman]

In my original posting
http://www.cs.nyu.edu/pipermail/fom/2006-February/009882.html

regarding the Gold Standard, I reiterated the de facto Gold Standard of ZFC,
and gave a short abbreviated list of some major stopping places in the
apparently canonical hierarchy of logical strengths. These levels ranged
from "barely even start" to "barely even stop".

There I basically challenged avowed predicativists to address this de facto
Gold Standard. E.g., reject it outright and recommend that it replaced, or
whatever.

Some chose to ignore the issue of what to do about the de facto current GOLD
STANDARD of ZFC. 

This gold standard of ZFC must be a very difficult pill to swallow for
predicativsts, as it constitutes a determination of no confidence in
predicativist restrictions, and an acceptance of the coherence and
naturality of ZFC. 

Actual responses from avowed predicativists were of two kinds.

1. Condemning impredicativity by reiterating the claim that "impredicativity
has no clear philosophical basis".

2. It doesn't make any difference anyways, since mathematicians do not use
impredicative arguments, or they do not use impredicative arguments for
"important" results by "important" people, or the impredicative methods for
the "important" results are easily removed.

With regard to 1. This is clearly a misuse of the word "philosophical", and
predicativists here on the FOM avoid directly criticizing the obvious "clear
philosophical basis" that has been accepted by the mathematical community.

There is no acknowledgement that predicativity itself is subject to the same
complaint from lower down - by finitists who may believe, with equal fervor,
that "the completed totality of all natural numbers has no clear
philosophical basis".

I find 2 above especially noteworthy, as it represents the really productive
challenge. 

In fact, the same thing has been said by finitists:

2'. It doesn't make any difference anyways, since mathematicians do not use
infinitary arguments, or they do not use infinitary arguments for
"important" results by "important" people, or the infinite methods for the
"important" results are easily removed.

There are important counterexamples to 2 and even more to 2'. The
counterexamples will grow to 2 and the counterexamples will grow to 2'.

Predicativists may well think, quite wrongly, that the counterexamples to 2
will not grow significantly, whereas the counterexamples to 2' will grow
significantly. 

As for ultrafinistism, I heard Andy Gleason assert (fancy core
mathematician), at an AMS meeting, something to the effect that

*it doesn't really make any difference if we prove our results for all
integers. We don't really lose anything by just proving them all reasonably
sized integers. That should be enough for us.*

So from the point of view of the ultrafinistist, the finitist looks wild,
working "without clear philosophical basis". From the point of view of the
finitist, the predicativist looks wild, working "without clear philosophical
basis". From the point of view of the predicativist, the normal
mathematician (working with lubs and maximal principles and Zornifications
and minimal bad sequences and Topoi and Grothendieck universes) looks wild,
working "without clear philosophical basis".

Harvey Friedman