[FOM] Foundationalist introduction to the field with one element, part I

[Colin McLarty]

On Sun, Jun 19, 2016 at 10:49 PM, Harvey Friedman <hmflogic at gmail.com>
wrote in response to my Sun, Jun 19, 2016 at 7:13 PM:

I get the impression that there is something misleading about
> referring to this enterprise as searching for the 1 element field.
>

I think it is meant as kind of a joke, both because we know there is no
one-element field, and because anything with just one element seems utterly
trivial—while this topic is far from trivial.

I say this because I think that there real problem you are talking
> about is: how should be overhaul the entire finite field concept in
> such a way that we have *-fields of exactly the cardinalities p^n, n
> >= 1, where p is a prime or 1, and the applications and connections we
> have for and reasons for worshipping the usual finite fields remain in
> tact for the *-fields.
>

Well, it is close to that.  But this is a algebraic geometers' project.
They aim to overhaul the concept of "algebraic variety over a field" rather
than the concept of field.  The various approaches to it actually do not
have any algebraic object that is the "one-element field."  Rather they
have well defined objects that are "varieties over the one-element field."

In the terms I'm using, each current approach defines P^n(F1), the
n-dimensional projective space over F1, but none actually define F1.

> From this point of view F1 should be a subfield of every finite field,
> which
> > is impossible with the standard definition of field.
>
> Presumably because of the divisibility law that if one finite field's
> cardinality divides another, then the former finite field is a
> subfield of the latter finite field?
>

This is my mistake.  The snippet quoted from me should have said the
spectrum of F1 is a *quotient* of each finite field.  That is impossible by
the standard definition of field for an even simpler reason of
divisibility: no field has any proper quotients since a field homomorphism
cannot kill any unit.  In Harvey's context of pseudo fields, and even in
the larger context of all rings, the one element ring is a quotient of all
other rings.  But it does not achieve that the one-element field is meant
to.

This correction, saying quotient instead of substructure, actually makes
Harvey's proposal even more germane.  I will not repeat it here but it is at

http://www.cs.nyu.edu/pipermail/fom/2016-June/019923.html

I have no strong feeling about that project.  It is intersting to me to
think about the contrast between Harvey's basically algebraic idea and the
geometric project begun by Tits.  For now I will try to write a brief
account of current geometric approaches.
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