Infinite versus unendlich and double-negation

Question

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).

Answer 1

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...

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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.