[FOM] A Defence of Set Theory as Foundations

[Nik Weaver]

Robert Lindauer wrote:

> Personally, I can't imagine the world where someone actually checks the
> twin prime conjecture by running through all of the integers and
> checking whether each was divisible, etc.
[snip]
> I simply don't know what to imagine when those kinds of things are
> said.  This is because I can't really imagine the process that would
> achieve such things, perhaps you could offer something more descriptive
> to help me imagine such a process?

Thank you for the opportunity to explain further.  As I am sure you
know, there is a fair amount of philosophical literature on "supertasks"
(or "hypercomputation"), and some rather explicit proposals have been
made as to how such computations could be performed.  A few authors
have even been so bold as to claim that infinite computations are
actually physically possible in our universe (e.g., by using quantum
fields or by jumping into a rotating black hole) --- though I haven't
yet been convinced by any such claim.  These proposals have been
sharply criticized, but I don't think anyone has claimed that they
are *logically* impossible, or *literally inconceivable*.

For my purposes the details of how one imagines performing an
infinite computation are irrelevant, as is the question of whether
there is really any practical possibility of doing so.  As I have
tried to emphasize, my concern is the realm of logical possibility.

(Please take the last paragraph seriously before responding with
comments about black holes, etc.)

If you really insist that infinitely long computations are logically
impossible, there may be little further for us to discuss.  I suppose
you would also reject the concept of the completed infinite in any
form.  I believe that is a coherent position, though I think it has
rather drastic consequences.

> Finally, I don't see why one has to imagine another world in order to
> recognize that 1 = 1.  1= 1 in our world, doesn't it?

I guess I would just ask you to reread whatever it was I wrote that
made you think I was claiming this.


Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu