Mathematics with the potential infinite
Vaughan Pratt
pratt at cs.stanford.edu
Sat Feb 4 19:58:47 EST 2023
Matthias makes a good point that there are paradoxes depending on
uncountability of the continuum, and I didn't mean to imply otherwise by my
recent remark that its uncountability was a bit of red herring when it came
to the possibility of paradoxical aspects of the continuum.
My first post here on this subject defined "the gapless geodesic" and
proved it was isomorphic to R. I then asked 1. whether that definition
involved any actual infinities, and 2. whether the uncountability of R
could be proved using only potential infinities.
In my most recent post (before this one), I partly addressed my first
question by using Zeno's paradox to quibble with the premise that
infinities can't be achieved in practice. A convergent series is feasible
when you also allow the time spent on the finite prefixes of the series to
converge, which resolves Zeno's paradox. Obviously the L's and U's in my
definition of a gapless geodesic have to converge to the gap separating
them and therefore aren't the sort of actual infinity that is infeasible.
And in more detail I gave a two-dimensional version of the paradox
involving the non-constructibility (with straightedge and compass) of pi,
namely that length 2 pi can be constructed by drawing a circular arc of
unit radius with the compass, which also produces area pi. I pointed out a
feasible way of taking the limit of a simple variant of Archimedes' method
of approximating pi with ever-better constructible approximations (his
regular polygons), namely by bending a suitably flexible rod of length 4
into a circle of radius 2/pi. (Or length 2 for radius of 1/pi, except that
length 4 is what you get when starting with X and Y as the unit vectors.)
In this post I partly address my second question by pointing out that what
I actually constructed was an injective homomorphism to the gapless
geodesic (G, +, 0, <=) from any Archimedean additive group of reals that
included the dyadic rationals. (In the definition I overlooked that
density did not ensure presence of all dyadic rationals; to fix this,
strengthen condition (i), density of <=, by including a second binary
operation, namely midpoint.) This is because the last step of the
construction only dealt with the dyadic irrationals of R, and my proof
tacitly assumed that those filled every gap in (G, +, 0, <=).
So if you're the sort of constructivist who prefers to take R to consist
only of the computable reals (defined to include the limits of computable
sequences), then the gapless geodesic will exceed R unless you restrict L
and U to computable sequences of dyadic rationals.
So rather than claiming to have proved that the gapless geodesic is
isomorphic to R, it would be better to claim that up to isomorphism there
is only one gapless geodesic. This works even if you impose some sort of
restriction on L and U such as computability, since that restriction will
be applied uniformly to all gapless geodesics.
And regardless of the choice of restriction the gapless geodesic must be
Archimedean because no restriction on L and U can eliminate the gap between
the finite and the infinite.
And (to answer my second question), if you *define* R to be the gapless
geodesic, it will only be uncountable if L and U are sufficiently
unrestricted, e.g. allowing all of Brouwer's "lawless" sequences, of dyadic
rationals in this case.
On a historical note, to the best of my knowledge the term "gap" as I've
defined it for a dense linear order first appeared (auf Deutsch of course)
in Hausdorff's *Mengenlehre* in Section 11 of Chapter 3, Order Types. If
you know of an earlier appearance I'm all ears. Hausdorff defines four
kinds of nontrivial ordered partitions of a linear order, the other three
being a "jump" witnessing nondensity and the two kinds of "cut" (in the
sense of Dedekind). He points out that a continuum can be defined as any
linear order that is free of jumps and gaps; my preference for including
group structure at the same time when defining "geodesic" is to efficiently
rule out the many "monsters" in the sense of Lakatos that otherwise emerge
unbidden.
Vaughan Pratt
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