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Is there a kind of "consensus" towards the meaning & intuition of the concept of "potential infinity" that goes back to Aristotle and is promoted by Edward Nelson, e.g. in the paper Hilbert's mistake?

Nelson distinguishes it strictly from "completed infinity", which is the kind of infinity mathematicians mostly refering to and which appears to be closer to what humans actually "intuitively" imagine under infinite entities, eg the naturals IN would be instance of an completely infinite object.

On the other hand, I have (as a mathematician) some troubles to develop an intuition for "potential infinity". Are there intuitional approaches to grasp this concept known?

Motivation: In comments below this answer Mikhail Katz explained that Nelson related the concept of potential infinity to concept of "numerals" - which in turn may be seen as a promising candidate to approach the concept of metalanguage integers.
Since originally my motivation was to develop an intuition for "metalanguage integers", I'm wondering if there is a philosophical consensus on how one may think of "potential infinity" before relating it to entities from (mathematical) metatheory.

Asking plainly, what is the precise contrast of potential infinity to "complete infinity" assuming one has say "school book intuition" for the latter?

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    You may want to edit your question to make it clearer that you don't think that Nelson introduced the concept of potential infinity, which is of course already in Aristotle as Nelson makes it clear in his text.
    – Mikhail Katz
    Commented Sep 18 at 16:08
  • Potential infinity was as old as Aristotle and there is no "Hilbert mistake".
    – Mauro ALLEGRANZA
    Commented Sep 18 at 16:54
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    And Predicative Arithmetic
    – Mauro ALLEGRANZA
    Commented Sep 18 at 17:29
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    I edited the post to remove references to Nelson's 'special role', he is just a recent popularizer of a very old concept. The mathematical version of potential infinity is the intuitionist/constructivist notion of it shared by all pre-Cantor mathematicians and formalized by Brouwer and Weyl. Infinite objects there can only be represented through constructive generation procedures that never 'finish'. For intuition, look at intuitionistic treatment of natural numbers, Brouwer's choice sequences and the real continuum in SEP, for example
    – Conifold
    Commented Sep 18 at 23:18
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    @Conifold, your comment conflates several types of projects in the philosophy of mathematics. For example, Hermann Weyl's project was predicativism (rather than intuitionism). Similarly, Nelson's type of finitism should not be conflated with Intuitionism, because the key feature of the latter is the rejection of the law of excluded middle, which was not a feature of Nelson's finitism. To get a better idea of these various approaches, I would recommend reading Abraham Robinson's essay Formalism 64 (published in 1965), where these approaches are discussed in detail. As far as the SEP ...
    – Mikhail Katz
    Commented Sep 19 at 11:33

2 Answers 2

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Quite simply, potential infinity can be thought of as natural numbers prior to a set-theorisation thereof. As you know, Peano Arithmetic (PA) is a theory that does not know of any infinite sets. Furthermore, PA is biinterpretable with the theory obtained from ZF by replacing the axiom of infinity by its negation (definitely no completed infinities here!), and adding epsilon-induction; see e.g., Caicedo's answer and Enayat's answer.

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  • I think that by "Hilbert's Mistake", the author is simply referring to the general surprise in the Mathematics community by Gödel's Incompleteness Theorems, which could alternately be interpreted as "Inconsistency Theorems" if one denies the incompleteness stuff.
    – Him
    Commented Oct 9 at 7:54
  • just to clarify, by "set-theorisation of blabla" where blabla should be some meaningful mathem object, you refer to routines in spirit of Gödel numbering encoding all mathem objects living so far in some fixed universe as numbers - or, in case of set-theoretisation - as sets, right?
    – user267839
    Commented Oct 9 at 9:01
  • No, I am simply referring to the set-theoretic presentation of N, built from the empty set, then the set containing the empty set, then the set containing the empty set and the set containing the empty set, etc. Namely the von Neumann "construction" of N. In such a presentation, N "exists" as a completed/actual infinite totality.
    – Mikhail Katz
    Commented Oct 9 at 9:04
  • And what means precisely to say that a theory T "not know of any infinite sets"? That it not contains infinity axiom (ie, forcing any model of it to contain an infinite set, like eg ZF does)? Or, that the formulas inside this theory are not allowed to use indexings over infinite sets?
    – user267839
    Commented Oct 9 at 9:17
  • The best way of viewing it is as I mentioned in my answer: PA is biinterpretable with the theory obtained from ZF by replacing the axiom of infinity by its negation (plus epsilon induction). Infinite sets are expressly forbidden! On the simplest level, the syntax of PA simply does not speak of infinite sets.
    – Mikhail Katz
    Commented Oct 9 at 9:19
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what is the precise contrast of potential infinity to "complete infinity?"

People want to think of Infinity as a reeeeally big integer that's larger than any other. That's not what's going on.

Let's make it a little simpler by making Infinity not so far away. Bring Infinity down to a finite value by using a hyperbolic metric like Escher used in his "Circle Limit" drawings:

The devils get smaller and smaller, but because there is no smallest devil, they never actually touch the outer "circle limit."

The answer to your question is, it's the difference between an open set inside a circle that reaches right to the edge but never touches it (because there is no smallest devil), and the closure of the that set which is the circle itself.

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  • Not sure if I got the right analogy, but from topological viewpoint isn't this the same difference as between non compact objects and their compactifications, compare eg with treatise of so called cusp points of modular curves (en.wikipedia.org/wiki/Modular_curve) where one for appropriate reasons prefer to work with compactified objects
    – user267839
    Commented Oct 10 at 10:31
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    From the point of view of traditional mathematics, there is a completed countable infinity of the bat figures already inside the open disk. There are no bats on the boundary at all. How would adding the boundary explain a transition from potential infinity to completed infinity?
    – Mikhail Katz
    Commented Oct 10 at 11:55
  • @MikhailKatz Well, I guess I don't understand the concept of "completed Infinity" -- if it means anything at all.
    – Miss Understands
    Commented Oct 10 at 18:48

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