Re: Point-free affine real line?

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

Sorry, what I wrote was a bit sloppy. Vaughan is right that the problem
doesn't arise with the passage from considering Q as a linear space
to considering it as an affine space, since it already has order-
reversing linear automorphisms. For the order to be definable from
the algebraic structure, you need to consider Q as a field, which is
what the usual Dedekind-section construction does.

Peter Johnstone

On Thu, 7 Jun 2018, Vaughan Pratt wrote:

> > 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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