[FOM] The Field with One Element?

[Colin McLarty]

Two different senses of "foundational" are being used here.

The one-element field is in no way an issue for ZFC foundations versus
categorical foundations or Homotopy foundations or anything like that.  It
is even easier from that point of view than Mario Carniero suggests.

The one element field is a matter of finite combinatorics and so it has
more to do with Subsystems of Second Order Arithmetic.

But the one-element field is a big challenge to the philosophically
foundational idea that mathematics deals with structured sets.

The point of the one-element field is not to put some structure on a set
which will generalize the field axioms so as to include a model with 0=1.
That is easy to do. Harvey's straightforward modification of the field
axioms achieves that.

Rather the point it is to set up not only a "field" where 0=1 but a notion
of algebra over this field such that (as Andrius Kulikaukas says) the
Gaussian binomial coefficients can be interpreted as counting the number of
k-dimensional subspaces of an n-dimensional vector space over a finite
field Fq, with q elements, even when q=1.

Very many facts of combinatorics can be expressed as algebra over finite
fields, and seem to naturally extend to  the case of one element.  But no
precise way of extending the ideas of field and vector space etc seems to
justify this case.  A straightforward model-theoretic extension of the
classical notions, to the generalized field axioms allowing 0=1, is easy to
do.  And it clearly does not give the right results for vector spaces etc.

Rather what is need is a whole framework of linear algebra with a
novel "combinatorial
interpretation" as Andrius says.  That challenge has been enticing ever
since Tits suggested it but it has not yet been done.

Colin

On Fri, Jun 17, 2016 at 10:29 PM, Harvey Friedman <hmflogic at gmail.com>
wrote:

> On Fri, Jun 17, 2016 at 2:41 PM, Andrius Kulikauskas <ms at ms.lt> wrote:
> > Dear Harvey,
> >
> > Thank you for your invitations in your letter below and also earlier,
> "...I
> > am trying to get a dialog going on the FOM and in these other forums as
> to
> > "what foundations of mathematics are, ought to be, and what purpose they
> > serve"."
> > http://www.cs.nyu.edu/pipermail/fom/2016-April/019724.html
> >
> .>..Would the "field with one element" be such
> > issue for you?
> > https://ncatlab.org/nlab/show/field+with+one+element.
>
> First tell me what is wrong with the following plan for the field with
> 1 element?
>
> DEFINITION. A pseudo field is a nonempty set F together with elements
> 0,1 (possibly the same) and binary functions +,dot, such that the
> following holds.
> 1. x+0 = x.
> 2. x+y = y+x.
> 3. x+(y+z) = (x+y)+z.
> 4. For all x, there exists y such that x+y = 0.
> 5. x dot 1 = x.
> 6. x dot y = y dot x.
> 7. x dot (y dot z) = (x dot y) dot z.
> 8. For all x not 0, there exists y such that x dot y = 1.
> 9. x dot (y+z) = x dot y + x dot z.
> 10. If x dot y = 0 then x = 0 or y = 0.
>
> THEOREM. There is a pseudo field with exactly one element.
>
> THEOREM. The pseudo fields with more than one element are exactly the
> fields.
>
> Proof: Suppose the pseudo field has 0,1 the same, and there is more
> than one element. Let u be other than 0,1.
>
> We now show that u dot 0 = 0. We have u dot (1 + 0) = u dot 1 = u = u
> dot 1 + u dot 0 = u + (u dot 0). In particular u = u + (u dot 0). Let
> u + v = v + u = 0. Then 0 = v + u = v + (u + (u dot 0)) = (v + u) + (u
> dot 0) = 0 + (u dot 0) = u dot 0. So u dot 0 = 0.
>
> But u dot 1 = u. Since 1 = 0, we have 0 = u. This is a contradiction. QED
>
> Harvey Friedman
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