Re: ​Re: Mathematics with the potential infinite

[Matthias]

In my opinion, both claims make sense from the perspective of a 
potential infinite. In more detail:

Define two types/classes: "real" (for real numbers), together with a 
constant 0 : real, a function * : real -> real -> real, a relation <= : 
real -> real -> bool (and probably more). And similarly a type 
"geodesic".

Both types have an increasing extension, finite at each stage. So the 
interpretation of * is: For each element a_0 (of type real) at stage 
i_0, and each a_1 (of type real) at stage i_1, there is an element a_0 * 
a_1 (of type real) at a sufficiently large stage i_2. At each of these 
stages there are only finitely many elements. This idea can be applied 
to the other constants and properties in the obvious way as well (e.g. 
density).

The upper and lower sets U : geodesic -> bool and L : geodesic -> bool 
are finite sets at each stage, depending on the stage of extension of 
the whole set "geodesic" (they are subsets of it). At each stage they 
have a greatest/least element. However, the statement that they have (as 
a whole, dynamic set) no greatest/least element is statisfied (e.g. for 
set U) whenever for each element a_0 in U (at stage i_0) there is an 
element a_1 (at some larger state i_1) such that a_1 < a_0.

The bijective map iso : geodesic -> real is an extensible set of 
assignemnts a |-> iso(a), finite at each stage.

Since the inference rules can remain valid, the proof should be no 
problem either.

A short comment on uncountablity (non-enumerability), which is somehow 
related to Brouwer's lawlike and lawless choice sequences: In the 
context of a potential infinite, no infinite set is exhausted 
completely, the difference between enumerable and non-enumerable sets is 
that for an enumerable set each element will be "hit" at some point by 
the recursive procedure. If the class is non-enumerable, this is not the 
case, since there is no procedure that creates all elements. However, no 
element in such a non-enumerable set will be "forgotten": Whatever 
element you can think of, it can be used as the next step in the always 
incomplete process.

Regards,
Matthias


------ Originalnachricht ------
Von "Vaughan Pratt" <pratt at cs.stanford.edu>
An fom at cs.nyu.edu
Datum 28.01.2023 09:13:40
Betreff ​Re: Mathematics with the potential infinite

>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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