​Re: Mathematics with the potential infinite

[Vaughan Pratt]

My apologies for not having previously followed threads on this topic.
However after seeing Stephen Simpson's message just now (Friday) it
occurred to me to ask whether an uncountable set could be described using
only potential infinities, for example the real numbers (R, *, 0, <=) as a
linearly ordered group under addition, compatibly ordered in the sense that
each of the group multiplication's two arguments is monotone: if x <= y
then x*z <= y*z, and likewise for the right argument.  (* = +.)

Define a *geodesic* to be a nondegenerate linearly ordered group (G, *, 0,
<=).  (Although G is not assumed abelian, the linear order makes it
abelian.)  Examples include the integers, the dyadic rationals, every field
between the rationals and the reals, and many non-Archimedean extensions
thereof.

Call a geodesic G *gapless* when (i) it is dense, and (ii) for every
nonempty suborder (U, <=) of (G, <=) having no least element, and every
nonempty suborder (L, <=) of (G, <=) with L < U and having no greatest
element, such that there is at most one element of G between L and U; then
there exists an element of G between L and U.

I claim that every gapless geodesic is isomorphic to R with the above
structure.

(Proof outline: Take any element x of G with 0 < x and pair 0 and x with 0
and 1 in R.  Pair the integers in R with the subgroup of G generated by x,
cyclic and therefore abelian.  Repeatedly divide the intervals in (n, n+1)
in G into two equal parts and pair the results with the dyadic rationals in
(0,1), a dense set.  Pair each dyadic irrational q in R with the unique x
given by the gaplessness condition for any L and U in G whose counterpart
in R converges to q from each side.  Lastly, G must be Archimedean or there
would be an empty gap between the finite and infinite elements of G.)

1.  Do these definitions, claims, and constructions meet the criteria for
only potential infinities?

2.  Can R be shown to be uncountable using only potential infinities?

(Those familiar with Otto Hoelder's 1901 paper showing that every
Archimedean linearly ordered group is isomorphic to some subgroup of R
under addition, which may be anywhere between Z and R, may see some
similarity of ideas in the above.)

Vaughan Pratt
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