# FOM: new axioms panel

Stephen G Simpson simpson at math.psu.edu
Thu Feb 17 16:38:12 EST 2000

```A response to John Steel's posting of Feb 16 2000.

John, THANK YOU VERY MUCH for getting into this discussion!  I love
the fact that we can ask questions of some of the panelists right here
on FOM.

> I greatly admire Harvey's work directed toward finding concrete,
> natural, combinatorial consequences of large cardinal hypotheses.

Me too.

> I might point out that if this work has long-term significance,
> then so do large cardinals,

Why do you think so?

1. The result that Harvey is looking for will say something like ``X
is equivalent to 1-consistency of Y'', where X is a mathematically
appealing finite combinatorial statement, and Y is a large
cardinal.  Thus X is independent of ZFC, but Y is not needed to
prove X -- only 1-consistency of Y is needed.  [ Unless we identify
existence of Y with 1-consistency of Y, as in my
existence-equals-consistency argument.  But nobody is going to
accept that argument, are they? ]  Therefore, couldn't it well turn
out that X and the independence of X from ZFC would have much more
long-term significance than Y itself?

2. Harvey also says he will be able to modify the above result to
something like ``W is equivalent to 1-consistency of ZFC'', where W
is similar to but more elegant than X.  So W is independent of ZFC,
but large cardinals are not involved at all.  Couldn't this
modified result also have a lot of long-term significance?

> whereas the converse is not true,

Hmmm.  ``The converse is not true.''  What does this mean?  Well,
according to the usual truth tables, not(converse(if-A-then-B)) means
B is true and A is false.  So John seems to be saying that large
cardinals have have long term significance and Harvey's work does not.
But this cannot be what John intended, because John greatly admires
Harvey's work.  So, what did John intend?

> therefore the long-term significance of this work is subject to at
> least as much doubt as is the long-term significance of large
> cardinals.

Why do you think so?  John, I hope you will elaborate.  The part I am
especially interested in is the Harvey aspect.  Assuming that Harvey
succeeds, what reason is there to doubt the long-term significance of
these results?

By the way, this kind of critical discussion is always welcome on FOM.
Isn't it, Harvey???  :-)

-- Steve

```