Mathematics with the potential infinite

[James Moody]

Matthias, I started reading [1] (have not started [2]), but I am wondering
about this section where you contrast your interpretation with other
interpretations of potential infinite:

> Recently Linnebo and Shapiro [5] suggested to formalize the potential
infinite using a modal reading of this notion. They rely on an analysis of
the potential infinite worked out by Niebergall [10].
> All of the mentioned approaches use some translation, restriction or
adoptions of the original axioms. Our approach is new, to our knowledge,
insofar as it formalizes the notion of an indefinitely large finite as a
relation, being part of the interpretation. As a consequence, no
manipulation of the axioms is necessary.

>From my perspective, your approach sounds similar to a modal approach with
transworld identity where the existential quantifier E.x is interpreted as
"possibly exists x", and the universal quantifier A.x is interpreted as
"forall x possibly".*** The usual semantic duality NOT E.x ~ A.x NOT does
not work the same way as in classical logic, since "not possibly exists x
phi(x)" entails but is not entailed by "forall x possibly not phi(x)". In
the degenerate case where there is only one possible world containing the
intended infinite model, however, the duality does hold (so it's a limit
phenomenon).

The notion of transworld identity would be just be given by implicit
embeddings of finite partial models in the directed system, and we also
assume all primitive relations between elements hold iff they hold
necessarily: if Wi embeds in Wj, and p1, ..., pn in Wi, then R(p1, ... ,
pn) in Wi iff R(p1, ..., pn) in Wj. Or expressed more modally, we add the
axiom "A iff necessarily A" for A atomic (atomic relations represent
"essential" properties / attributes which cannot vary across worlds).

There is obviously the subtle issue of functions, but assuming we are OK
representing them as relations with axioms saying "I am the graph of a
function", then for example the usual axiom schema for induction in PA is
interpreted (without any further translation required), to the following:

Forall y possibly (
    if (
        phi(0, y)
        and forall x possibly (
             if phi(x, y) then
             possibly exists s (
                 Succ(x, s) and phi(s, y)
             )
        )
    then forall x possibly phi(x, y)
)

For worlds Wi := {0, 1, 2, ..., i-1} (and obvious accessibility relation
turning this into a directed system), and simplifying to the case there are
no parameters to the formula phi being inducted (for simplicity), this ends
up being interpreted as:

if phi(0) and forall x, in some bigger world (
    if phi(x) then in some even bigger world (
        exists s Succ(x, s) and phi(s)
    )
)
then forall x, in some bigger world phi(x)

Thinking like Skolem, the "bigger world" here just needs to provide
witnesses to the relevant existential statements that might appear in phi.
It collapses to the usual interpretation of PA axioms when you take the
directed system of possible worlds to be the singleton {N}.

For nested quantifiers, e.g. totality of successor function "A.x E.y
Succ(x,y)", there is a another way of reading it as potentialist statement
about constructing mathematical objects:  "for any natural number x so far
constructed, we can construct new natural numbers so that there exists a
successor to x". Then there are subtle issues that an ultrafinitist might
be concerned about like whether possible worlds should be allowed to have
gaps or not.

Is there a difference between this modal explanation of potential infinite
and your presentation? I believe this modal explanation would count as an
"interpretation" and not just a "translation", since (aside from the
subtlety about functions), it just provides a new modal interpretation to
universal and existential quantification. I am wondering if this is
equivalent to your presentation, since your notion of "context" sounds
similar to how you might track the semantics of nested modal statements
that quantify over elements of other possible worlds.

Another somewhat related question: requiring the finite partial models /
worlds to be a directed system seems to me like a condition that avoids a
lot of the troubles that Saul Kripke worked on with transworld identity,
e.g. you don't have to consider "If in some possible world I had an
identical twin, would I be the same as the twin on the left or the twin on
the right?", since there is a unique way to embed you (or any particular
element of your current world) into any world visible from your world. You
could also weaken this condition, however, and end up with a kind of
multiverse, with multiple possible completions to an infinite model. In
that case, I would guess that the "true" sentences would be those that are
true in all of the possible infinite completions that could be obtained
from choosing a particular directed subsystem? Have you explored this?

Best,
James Moody

N.B. (***) For a universe U of (finite) worlds, phi is true in U if phi is
true at every world in U. This (together with the worlds forming a directed
system) explains why we don't use "necessarily" modality to describe
universal quantification.

On Tue, Jan 24, 2023, 5:59 PM Matthias <matthias.eberl at mail.de> wrote:

> 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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