Mathematics with the potential infinite
James Moody
jmsmdy at gmail.com
Sun Feb 5 13:28:01 EST 2023
Vaughan, regarding your questions 1 and 2, it would be important to
remember that "uncountability" is defined as the non-existence of a map
satisfying certain properties, and here it is important how we interpret
"map". In mathematics based on ZF(C), "map" is taken to mean a set of
ordered pairs, and a finitist / potentialist would presumably deny that any
actual infinite set of ordered pairs exists, even just the obvious
"identity map" on the natural numbers {(0,0), (1,1), (2,2), (3,3), ...}.
An interpretation of "map" that is more amenable to representation in
finitist / potentialist mathematics would be "procedure". In that case, the
trivial assertion "the natural numbers are countable" will come out as true
for a potentialist, because there exists (a finite description of) a
procedure P and (a finite description of) an inverse procedure Q so that
necessarily for all n P(Q(n))=n.
I think there is significant room for diversity in what counts as a (finite
description of a) procedure for a finitist / potentialist. One approach
that could allow a potentialist to express your "there is a unique gapless
geodesic" statement would be to represent real numbers as (finite
descriptions of) procedures which take natural numbers and produce (dyadic
rational?) approximations. Then an injection from some linear order M to G
could be (a finite description of) a procedure P which takes (a finite
description of) procedure p describing a member of M, and produces (a
finite description of) a procedure q describing a member of G, together
with an "inverse" procedure Q, so that necessarily for all p in M,
Q(P(p))=p.
The subtlety here (which this definition accommodates) is that on some
accounts distinct procedures may define "the same" real number. On this
definition, the Q(P(p)) is required to be literally equal to p (as a
description), which is strictly stronger than both "equality in the limit"
or "indistinguishable by finite means". Do you think your argument for
"uniqueness of G" would go through with this very strict notion of
"injective map"? The point here would be to avoid altogether the
problematic notion of "limit".
In any case, the usual "diagonalization" arguments for showing
uncountability tend to be effective. So I imagine there would be no problem
constructing a procedure D which takes any putative procedure claiming to
give a surjection Q : N -> G and produces an element q := D(Q) of G for
which "necessarily not exists p with Q(p) = q". A potentialist would
presumably accept the existence of the diagonalization procedure D, because
you can write it down (albeit it requires quite a few pages)! The existence
of D then implies it is impossible for a surjection procedure Q : N -> G to
exist, so by the usual definition, suitably re-expressed to the
satisfaction of a potentialist, G is "uncountable". None of this requires
accepting actual infinities, although it does require accepting the
existence of procedures and other "higher level" (meta)mathematical
objects.
James
On Sun, Feb 5, 2023, 10:05 AM Vaughan Pratt <pratt at cs.stanford.edu> wrote:
> 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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