The German term for infinite is unendlich, which transliterates as non-ending, or non-finite.

This is just word-play but from a constructive point of view, is the shift from a negative to a positive concept of infinity consequential? (one could similarly contrast, eg, discrete versus non-continuous or other dichotomies).

Concretely, is there a logical distinction between the proposition "$p$ is not-infinite" versus "$p$ is not non-finite?"

It would seem that the latter, but not the former, is a pseudo-truth, as defined by Kolmogorov in his paper on the principle of excluded middle (EM) as the double negation of a proposition.

Motivation for considering this comes from basic observations:

  • Both set theory and topos theory typically include an explicit axiom of infinity.

  • In Moschovakis' Descriptive Set Theory the axiom of Infinity is defined in terms of the existence of a set that contains the empty set and the union $x \cup \{x\}$ for any $x$ in this set.

Moschovakis' definition clearly generates an unending diversity of nested sets. However is that the same thing as an actual infinity (constructively, this process would never end).

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    $\begingroup$ "Infinite" also translates to non-ending or non-finite, since "in-" is a negating prefix in Latin. $\endgroup$ – Henning Makholm Jan 3 '13 at 13:44
  • $\begingroup$ Good point. I'm aware of that (I grew up in Italy) yet somehow the negation seems internalized in "infinity" as a sort of habituation. $\endgroup$ – alancalvitti Jan 3 '13 at 13:48
  • $\begingroup$ I'd like to add that the correct uninflected form of the adjective would be "unendlich" rather than "unendliche". $\endgroup$ – Andy Brandi Jan 3 '13 at 15:06
  • $\begingroup$ @AndyBrandi, thanks - edited. $\endgroup$ – alancalvitti Jan 3 '13 at 16:00
  • $\begingroup$ @HenningMakholm, ok so there's no difference between the Latin and German in terms of the negation. But doesn't the question remain as to the constructive validity of ~infinite = ~~finite, which is not constructively equivalent to finite without the Excluded Middle principle. $\endgroup$ – alancalvitti Jan 8 '13 at 0:46

If you are willing to accept that:

  1. Sets cannot be both finite and infinite.
  2. Something is infinite, if and only if it is not finite.
  3. Everything is either finite or infinite.

Then to say that $p$ is not-infinite is the same saying $p$ is finite and it is the same as saying that $p$ is not non-finite.

Philosophically speaking, we can ask what is finite? Does finite mean equinumerous to a finite ordinal? Without the axiom of choice one has to make the distinction $|A|\nless\aleph_0$ vs. $\aleph_0\nleq |A|$. This gives a little space for a "philosophical" debate for what is infinite.

Do we define infinite sets as not-finite, or do we require them to have some combinatorial property (such as a bijection with a proper subset; or even just a surjection from a proper subset; and more). Regardless to how you define it there are two things which seem indisputable:

  1. The finite ordinals are finite, no matter how you choose to define finiteness.
  2. Anything which contains a countably infinite subset is infinite.

As for constructive infinite, I heard they are still constructing that...

I want to add a tangential story. Yesterday on my way back from Jerusalem I met someone which is a hobbyist mathematician and a finitist. He believes there is meaning to the phrase "all the natural numbers" or "all the even numbers", but there are no such sets.

We discussed that it's easier to express infiniteness. You just need to assert that something is infinite. However to show something is finite you need to actually bound it. In this aspect there is one trivial schema of first-order axioms which assures infiniteness (add the sentence "There are $n$ distinct elements", for every $n$); but there is no such schema which assures finiteness.

  • $\begingroup$ Asaf, are your initial points 1. 3. true without EM and also the negation of both sides of 2? $\endgroup$ – alancalvitti Jan 3 '13 at 14:11
  • $\begingroup$ I think Dedekind wrote in an unpublished letter (I'd like that reference) something to the effect that the bijection to a proper subset is the only rational definition of infinite. (Elsewhere he also credits that definition to Bolzano and Cantor.). My question there is, when is the proper part constructively decidable? Eg, in Collatz conjecture: is the basin of attraction of $(4,2,1)^\infty$ cycle a proper part of the naturals? $\endgroup$ – alancalvitti Jan 3 '13 at 14:14
  • $\begingroup$ I don't know what is "true". In what sort of context? As for the definition of infinite, I find that if a set can embed all the finite ordinals then it must be infinite. Even if it cannot embed them all uniformly. In this aspect I much prefer Tarski's definition of finite to that of Dedekind: $A$ is finite if and only if every non-empty collection of subsets of $A$ contains a $\subseteq$-maximal element. $\endgroup$ – Asaf Karagila Jan 3 '13 at 14:21
  • $\begingroup$ What's the relation of Tarski's definition to Dedekind's? Ps, are you familiar w/ this paper "Independence of various definitions of finiteness" by Levi: matwbn.icm.edu.pl/ksiazki/fm/fm46/fm4611.pdf $\endgroup$ – alancalvitti Jan 5 '13 at 1:25
  • $\begingroup$ Tarski's equivalent to "true" finiteness without choice, whereas Dedekind's definition requires some choice. If you replace $\subseteq$-family by $\subseteq$-chain then you need some choice, but less than the amount required to show that every Dedekind-finite is finite. $\endgroup$ – Asaf Karagila Jan 5 '13 at 1:30

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