[FOM] Real Division Algebras + Tarski/Lefschetz Principles (typo fixed)
Adam Epstein
adame at maths.warwick.ac.uk
Tue Sep 10 14:28:53 EDT 2002
---------- Forwarded message ----------
Date: Tue, 10 Sep 2002 14:03:09 +0100 (BST)
From: Adam Epstein <adame at maths.warwick.ac.uk>
To: fom at cs.nyu.edu
Subject: Real Division Algebras + Tarski/Lefschetz Principles
First, here is Joe Shipman's response to my question about CH and the Law
of the Excluded Middle:
On Mon, 9 Sep 2002 JoeShipman at aol.com wrote:
> Here's an even more striking way of looking at it:
>
> All that you really need is "Every uncountable Borel set has cardinality
> continuum".
>
> Then CH implies every non-Borel set has cardinality aleph-one
> and not-CH implies every set of cardinality aleph-one is non-Borel.
>
> So you have two sets of sets of reals, "non-Borel sets" and "sets of
> cardinality aleph-one", one of which is strictly contained in the other,
> but we don't know which!
>
> Choice is essential here, you need AC both to get a non-Borel set and to
> get a set of reals of cardinality aleph-one.
>
> Instead of "Borel", you could use any other property of sets of reals
> that is satisfied by no sets of cardinality strictly between aleph-zero and
> continuum.
>
> -- JS
Turning to other matters, consider the theorem that any finite dimensional
real division algebra has dimension 1,2,4 or 8. [I think this has been
discussed somewhere on FOM, but not in detail].
Since this is true over the reals, it is also true over any real closed
field. Of course, what is guaranteed here is merely that for each n there
exists such a proof for the statement
S_n="If n is not 1,2,4 or 8 then there is no such algebra of dimension n".
If we could somehow insert a \forall n quantifier in front, it could be
convincingly argued that there is a "uniform" elementary proof. For
example, if one set up a (minimal) theory which contains the theory of
real closed fields AND Peano Arithmetic then one could presumably easily
formalize the statement \forall n S_n. In the parallel case of
algebraically closed fields, this could be done via resultants, so that
the quantified statement would amount to the assertion that all such
resultants are nonvanishing. Here I suppose it would require a bit more
work, but surely an analogous translation is possible.
One could then ask whether the appropriate translation of the statement
\forall n S_n - which would now be a sentence in the language of
arithmetic - is provable in PA (e.g. if one could inductively show that
the relevant "resultants" are all nonvanishing). This does NOT appear to
follow from the Tarski/Lefschetz Principle alone. In fact, the existence
of Ackerman-type phenomena in algebraic geometry (thread 71) make me very
doubtful that there should be any general guarantee that this can be done.
So two questions here:
1) Is this set-up (PA + RCF or PA + ACF) something which has been
thought through as a criterion for "elementary proofs"?
2) Is there (or need there be) such a proof of the 1,2,4,8 Theorem?
One could start by looking at the subtheorems for commutative division
algebras (1,2) and associative division algebras (1,2,4).
Hopf's proof of the (1,2) theorem uses topology, but not very much - just
the fact the n-dimensional sphere and n-dimensional projective space are
not homeomorphic when n>1. Maybe there is an elementary way to formulate
this consideration. I know that Classical Analysis is conservative over
PA, and it's reasonable to imagine that the statement that these two
explicit manifolds are not homeomorphic is/can be formulated and proved
"classically", though I'm not familiar enough with this criterion to
verify it myself. Furthermore, there does appear to a different and
"purely algebraic" proof, via Bezout's Theorem [in the 50's, by
Springer?]. This proof is often dismissed as unreadable/unenlightening,
but when I examined it years ago it seemed like it might qualify.
Frobenius' proof of the (1,2,4) theorem is quite evidently purely
algebraic, as is the later extension (1,2,4,8) to alternative division
algebras.
By contrast, the proof of the 1,2,4,8 theorem makes use of the apparatus
of K-theory. If there's a way to extract an "elementary kernel" I wouldn't
know what it is. (Offhand, I should know, but I don't, if this theorem
actually states that the only 8 dimensional algebra is given by the
Octonions, or if this can only be asserted for alternating algebras).
Adam Epstein
Mathematics Institute
Warwick University
Coventry CV47AL
United Kingdom
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