Re: Point-free affine real line?

[Vaughan Pratt <pratt@cs.stanford.edu>]

  > 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.

Morphism-wise, since the affine transformations are just the composition of
a linear transformation with a translation, and translation of the rational
line preserves order, affinity can't be the problem here.

Structure-wise, one can equip the rational line with either its linear
combinations or its linear order, or both.  Using both eliminates the
order-reversing linear transformations.   "Affine" only makes sense in the
context of having the linear combinations, as "affine" limits the linear
combinations to those whose coefficients sum to one.   If it is ok for the
linear combinations and the linear order to coexist, it must be even more
ok for the affine combinations and the linear order to coexist.

So whether one considers the morphisms or the structure they preserve,
affinity (affineness?) must be a red herring here: any problem for the
rational line as an affine space is surely also a problem for it as a
vector space.

Vaughan Pratt



On Tue, Jun 5, 2018 at 3:55 AM, Peter Johnstone <ptj@dpmms.cam.ac.uk> wrote:

> 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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