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

[Colin McLarty]

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).

Let q be any positive power of a prime, q=p^n, n>0.  Then up to isomorphism
there is a unique field Fq with q elements, and every finite (conventional)
field is of this form.  This has been an extremely productive idea in
algebraic geometry and combinatorics the past 70 years.

>From this point of view the one-element field F1 is the special case n=0.

>From this point of view F1 should be a subfield of every finite field,
which is impossible with the standard definition of field.  And further
considerations suggest F1 should be a subfield of the ring of integers Z,
which is also impossible by standard definitions.  To avoid overloading the
notation I will not use the expression p^n below, but much of the
motivation is that 1 is the special case of a prime power p^n where n=0.

As to the logical foundations of the idea: A lot of work on the idea of a
one-element field F1 considers only finite fields as motivation, and that
is all strictly finite combinatorics.  Considering the ring Z of integers
of course is still finitary in some way but it is not just finite
combinatorics.  Further considerations can link F1 to the complex numbers
and more, so the logical level of the idea of F1 will depend on things that
are not entirely settled in the literature today.

The issue for this post is to count the points in the n-dimensional
projective space P^n(F1) over  F1.

For any conventional field k the  points of the n-dimensional projective
space
P^n(k) over  k are equivalence classes of non-zero points in the n+1
dimensional vector space k^(n+1) where two points of k^(n+1) are treated as
identical when they are scalar multiples of each other.  If k is a finite
field Fq then the number of points is q^(n+1)-1 divided by q-1.  That is
just the number of non-zero points in k^(n+1), divided by the number of
scalars in k.

When k is a conventional finite field Fq this quotient can be written as a
finite sum q^n+q^(n-1)+...+1.

But when q=1 the quotient is 0/0, while the sum is simply n. The sum and
the quotient do not agree.

A correct treatment of the one-element field  F1 should make the number of
points of P^n(F1) equal n.  (More precisely, experts agree this should be
so, but the point of this discussion is that there is not yet any consensus
on precisely how to define the one-element field.)

But if we define the one-element field as the one-element pseudo field by
Harvey's axioms, and construct the projective spaces P^n(F1) as the usual
equivalence classes of non-zero points in the vector space F1^(n+1) then
 P^n(F1) has no points, since  F1^(n+1)  has no non-zero points.

Many more sophisticated combinatoric questions can be taken as
algebraic-geometric questions over finite fields Fq for q a positive power
of a prime.  Often one approach to such a question has a naturally
appealing, and significant, extension for q=1.  In our example the approach
to counting points of P^n(Fq) as a sum has an attractive generalization to
q=1.  But other approaches which are equivalent in the conventional case of
q>1 are not equivalent when q=1.  In our example counting points
of  P^n(Fq) as a quotient is undefined when q=1.

And in these more sophisticated questions, as in our example here, we can
make the case q=1 well defined by using the pseudo field axioms.  But that
well defined answer is not the intuitively correct answer in our example --
according to the intuition of people like Tits.  And the same is likely to
happen for other issues besides counting points of projective spaces.

Colin
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