[FOM] Number theory proof mentioned by Frege

John Baldwin jbaldwin at uic.edu
Thu Apr 17 00:14:15 EDT 2008

sheer speculation since I don't know the use of the words.

It is well known that if you try to have division algebras over the reals

only the complexes are commutative.

the quaternions lose commutativity, and with more losses you get degree 8 
and I think 16.  Then no more.

On Wed, 16 Apr 2008, Chris Gray wrote:

> In "The Foundations of Arithmetic", Frege mentions a proof from pp.106-7
> of Hermann Henkel's Theorie der complexen Zahlensysteme.
> "Hankel proves that any closed field of complex numbers of higher order
> than the ordinary, if made subject to all the laws of addition and
> multiplication, contains a contradiction."  Frege p. 106
> I have no copy of the Henkel book available to me.  Is this a well-known
> result?  Is this proof discussed or given elsewhere?
> Thanks,
> Chris Gray
> University of Waterloo Library
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