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

[Harvey Friedman]

On Sun, Jun 19, 2016 at 7:13 PM, Colin McLarty <colin.mclarty at case.edu> wrote:
> I will describe one typical but very limited issue about the one-element
> field F1 that is fairly easy to see, and I will compare Harvey's pseudo
> fields on these issues.  In a following post I will describe a more general
> issue which may be the key issue but is naturally more arcane.  That more
> general issue connects the one-element field to modern methods in math where
> the point is not to look at structures on sets but to look at structural
> relations between objects (even if the logical foundation is taken to be ZFC
> so that every object is actually a set).

I made a distinction in rough terms in my
http://www.cs.nyu.edu/pipermail/fom/2016-June/019920.html between
conceptual development of mathematical theories and f.o.m. At the
moment this still seems to be to be clearly in the realm of the former
and not of the latter. However, I am of course interested some in the
development of mathematical theories, particularly when the purpose is
intelligible or can become intelligible and exciting to a large
audience. Also, I am even more interested in general principles behind
the choice and development of a "good" mathematical theory.

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

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.

> 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?

The substructure condition, that the 1 element gadget is to be a
substructure of all the rest, seems to indicate that we must have to
play with equality here. First, let us suppose that we are not going
to play with equality. Then in any one element structure, we have
.that every operation maps everything to itself, and therefore must
map everything to itself in the remaining non 1 element structures.
This seems preposterous.

So we play with equality. There is an obvious commonly accepted way to
do so in normal mathematic, namely to replace equity with an
equivalence relation.

So this makes sense to me as a context to explore possible overalls of
the finite field concept to accommodate what you are asking for.

Namely, the field*'s are simply conventional fields together with an
equivalence relation on its elements. The cardinality of a field* is
simply the number of equivalence classes under its equivalence
relation.The 0,1 elements can be required to be distinct, but we don't
want to require that they be inequivalent. The field with one element
now becomes the *-field with exactly one equivalence class.

Now we certainly DO want to have some laws which involve the
equivalence relation with the other traditional components, 0,1,+,x,.

Now I think we may be in a position to enumerate some of the most
rudimentary conditions you want to impose and ask if there is, e.g., a
finitely axiomatized quantifier free or better theory which meets
these conditions. Maybe prove that this is impossible where the
demands are rather minimal. I wonder if impossibility results can be
obtained just looking at rather elementary conditions.

Harvey Friedman