Re: Point-free affine real line?

[Peter Johnstone <ptj@dpmms.cam.ac.uk>]

On Sun, 3 Jun 2018, Vaughan Pratt wrote:

> > The question about the affine real line represents a challenge to this
>> geometric approach, and I'd like to form a better idea of whether it is
>> simply a difficult problem, or a fundamental limitation to my approach.
>
> An affine space over any given field differs *k* from a vector space over
> *k* only in its algebraic structure, not its topological structure.
> Whereas the algebraic operations of a vector space over *k* consist of all
> finitary linear combinations with coefficients drawn from *k*, those of an
> affine space consist of the subset of those combinations whose coefficients
> sum to unity, the barycentric combinations.  Since the former includes the
> constant 0 as a linear combination while the latter does not, a consequence
> is that 0 is a fixpoint of linear transformations but not of affine
> transformations, whence the latter can include the translations.
>
> This is equally true whether *k* is the rationals or the reals.   So
> whatever method you use to obtain the real line from the rational line
> should also produce the affine real line from the affine rational line.
>
Not quite: the affine rational line doesn't have a definable total order,
since it has order-reversing automorphisms, so any definition using
Dedekind sections is problematic. However, it does have a ternary
`betweenness' relation, and it should be possible to rewrite the
geometric theory of Dedekind sections of Q, as presented on p. 1015
of `Sketches of an Elephant', in terms of this relation (but note that
sections will have to be unordered rather than ordered pairs of
subobjects of Q).

Peter Johnstone


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