Re: ​Re: Mathematics with the potential infinite

[Haim Gaifman]

Dear Vaughan,
Long time no hear no see, and it is very nice to hear from you.
The restriction of subscribing only to potential infinities (which can be traced back to Aristoteles) is Hilbert’s so called finitist position; Abraham Robinson agrees with him. Only structures based on proper initial segments of the natural numbers: {0, 1, 2,…, m} are accepted as legitimate, but for every m, if n > m, one accepts also the extension based on {0, 1, …., m, m+1,…., n}. The functions and/or relations that come with these structures are the usual functions and/or relations of PA (Peano Arithmetic). Of course, the functions are partial functions  , given the restrictions on the domain. 

PA, which is based on the standard model N of natural numbers, is much much… stronger than the theories
that arise within the framework of potential infinity.
One such interesting theory has been proposed by Skolem
and is known as PRA for Primitive Recursive Arithmetic. 

Now your question, if I understand you correctly, asks for a way of describing an uncountable structure using only potential infinities.This would be impossible, unless you allow countable non-standard model for the theory linearly ordered groups.

Best, Haim Gaifman

> On Jan 28, 2023, at 3:13 AM, Vaughan Pratt <pratt at cs.stanford.edu> wrote:
> 
> 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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