Mathematics with the potential infinite

[Matthias]

Dear FOM members,

for several years I have been engaged in developing mathematics solely 
with the concept of the potential infinity. Two publications of mine are 
now available.

The main idea is to see infinity as an indefinitely extensible finite 
and to develop a model theory that uses no (actual) infinite 
sets/objects at all. Nevertheless, the usual (classical, 
intuitionistic...) reasoning should be allowed. This is possible for 
first-order logic, for higher-order logic there are some particularities 
(see below).

A consequence of using the potential infinite in this way is, for 
instance, that the completed set N of all natural numbers does not 
exist. On a syntactic level, notations such as "n \in N" can be easily 
replaced by a type theoretic notion "n : nat". For the semantic side the 
main idea is that it suffices to have indefinitely large initial parts 
N_i := {0,...,i-1} of N, such that there is always a part N_i (= set N 
at stage i) that behaves exactly as N in the current context of 
investigation. So infinite sets are given by indefinitely (or 
sufficiently) large finite sets, and what "sufficiently large" means, 
depends on the context of the investigation. The context thereby 
includes stages of the syntax as well as stages of the model.

Developing this approach for first-order logic is possible, provided one 
interprets the universal quantifier with an implicit reflection 
principle, see [1]. Such an interpretation is necessary, since a naive 
interpretation of the universal quantifier by "for all ..." is 
equivalent to assume an (actual) infinite domain of objects. This 
approach goes back to Jan Mycielski and uses the technique of the 
Löwenheim-Skolem theorem (or Levy-Montague reflection principle). There 
are no technical difficulties for first-order logic. But since one 
cannot use set theory with actual infinite sets, one has to use 
higher-order logic in order to talk about objects such as functions, 
function spaces or the power set. The main idea here is to consider 
systems, i.e., direct systems, inverse systems and generalizations 
thereof, see [2].

For higher-order logic there are some challenges and differences to 
common practice. A difference is that one cannot equate first-order 
objects with higher-order objects, even if they are structurally 
isomorphic. So the set of abstractly defined real numbers R are not the 
same as (equivalence classes of) Cauchy sequences, or Dedekind cuts etc. 
An abstractly defined real number, e.g. Euler's number e, is a single 
object, whereas the Cauchy sequence with limit e is an unending process, 
not a single entity. One needs some mapping in order to relate them, 
e.g. lim(s) for a Cauchy sequence s.

Each function f : R -> R or f : CS -> CS (CS for Cauchy sequences) is 
allowed, also discontinuous functions such as f(x)=0 if x<0 and f(x)=1 
otherwise. Nevertheless, in the model all functions are continuous since 
an infinite sequence is treated as a potential infinite sequence and the 
function space is extensible as well. W.r.t. a function f : CS -> CS one 
might say that Brouwer's continuity principle is satisfied in the model 
theory, but it is not necessary to formulate this in the language, allow 
only continuous functions or use intuitionistic logic.

The difference of the set theoretic and this approach become also 
visible in the power set construction P(_), e.g. on the set of natural 
numbers N. If we understand infinite as potential infinite, the set N is 
seen as a direct system, P(N) as an inverse system, and P(P(N)) as a 
direct system again. This follows from the construction in [2]. A direct 
system is "finitary" in the sense that each element therein occurs at 
some stage (and then remains more or less the same), whereas an inverse 
system has "infinite elements" --- an element a is given only by 
approximations via the projections of a. So different to the power set 
construction in a set universe V, where P(P(N)) is "even more infinite" 
than P(N), the construction P(P(N)) based on the potential infinite is 
finitary again. The reason is (very roughly and informal) that if a 
subset A of N is given only by some state A_i and never as a completed 
set, then a property/set on A has always to deal with an unknown part at 
the end (the "open future development") of A, so it cannot itself 
develop arbitrarily, but must be defined by this very part A_i.

One of the main restrictions compared to mathematics with actual 
infinite sets is that totality of applying a higher-order functional to 
some function does not hold *in general* in the model. Assume that a 
functional F : (N -> N) -> N and a function f : N -> N are given. Then 
F(f) is not defined in general, only if one can show that this is the 
case, e.g., by proving that for all f there exists a value n such that 
F(f)=n. Technically, the reason has to do with the existence of filters 
on index sets in the model (see [2]).

Any feedback is warmly welcome.


[1] A Model Theory for the Potential Infinite. Reports on Mathematical 
Logic 57, 3-30, 2022.
https://rml.tcs.uj.edu.pl/rml-57/01-eberl.pdf

[2] Higher-Order Concepts for the Potential Infinite. Theoretical 
Computer Science, Volume 945, 2023.
https://authors.elsevier.com/a/1gMxJ15DaIAtMa


Kind regards,
Matthias
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