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To quote Kant (as usual!):

A quantity is infinite, if a greater than itself cannot possibly exist. The quantity is measured by the number of given units- which are taken as a standard—contained in it. Now no number can be the greatest, because one or more units can always be added. It follows that an infinite given quantity, consequently an infinite world (both as regards time and extension) is impossible. It is, therefore, limited in both respects. In this manner I might have conducted my proof; but the conception given in it does not agree with the true conception of an infinite whole. In this there is no representation of its quantity, it is not said how large it is; consequently its conception is not the conception of a maximum. We cogitate in it merely its relation to an arbitrarily assumed unit, in relation to which it is greater than any number. Now, just as the unit which is taken is greater or smaller, the infinite will be greater or smaller; but the infinity, which consists merely in the relation to this given unit, must remain always the same, although the absolute quantity of the whole is not thereby cognized.

The true (transcendental) conception of infinity is: that the successive synthesis of unity in the measurement of a given quantum can never be completed.

Without agreeing too strictly, I will say that Kant did notice the difference between a closed and an open infinity (my use of those terms "closed"/"open" is not the same as the mainline one, I will note). This is not the same as the difference between an actual and a potential infinity, though there is some overlap (there are actual infinities that are open, in that they already have infinitely many parts or elements but are open to more being added unto them; and there are closed potentially infinite sets, where the closure is given via the kind of parts/elements that can be added).

Now, I've been trying to find attempts at a solution to "the" Yablo paradox, but all I've found are debates over whether it is based on self-reference.❦ So instead, I'm pretty sure you can convert it into two different self-referential structures, and the whole Yablo set (hereafter yeti) does seem to self-refer via the Sn, n+1 schematic, "no matter what." More significantly, it seems like you can evolve the yeti into an ∃-yeti or an ∀-yeti, for S a sentence:

  1. S: ∃x((x = S+) → ⊥(x)) {"The sentence after this one is false"}
  2. S: ∀x((x > S) → ⊥(x)) {"Every sentence above this one is false"}

But so is the infinity of the yeti closed or open? If it is open, then next/higher sentences can be added in, but until they are added in, they do not "exist," and so whatever is currently the last subyeti is false (by presupposing next/higher subyetis). But the closed yetis also turn out to have an ordinally last subyeti, which is necessarily false (since this species of yeti is not allowed to eat any more sentences at all) (any transfinite ordinal will technically do, even successor ordinals (just stipulate that the set is closed at exactly that ordinal length), but if your logical stomach is grumbling, just use ORD if you have to). At any rate, the ∃-yeti turns out to have a different solution than the ∀-yeti:

  1. Since the last ∃-subyeti is false (because there is no next subyeti), then the 2nd-to-last is true. The 3rd-to-last is then false, the 4th-to-last is true, and so on.
  2. Since the last ∀-subyeti is false (for the same reason as in (3)), then the 2nd-to-last is true. Moreover, since there is one true subyeti above the 3rd-to-last, then the 3rd-to-last is false. But so then is the 4th-to-last false, etc. going all the way back to the zeroth/first.

My question is: what if the yeti is understood cardinally? So we say:

  1. S: ∃x((x = S') → ⊥(x)) {"There is at least one sentence (in the same set as this one is in), other than this one, that is false"}
  2. S: ∀x((x = S') → ⊥(x)) {"Every sentence other than this one is false"}

My intuition is telling me that if the cardinality of a yeti were not well-ordered, then reference to some "last" sentence that resolves the entire set does not as clearly go through. In fact, though, if such would not go through, then we would have a fairly weird derivation of the well-ordering lemma ("if ~WOL, then possibly an unsolvable yeti; ~an unsolvable yeti; therefore..."), maybe. However the issue is otherwise phrased, I will phrase it as: does the cardinal yeti seem to generate a paradox anyway (or, is it reducible to the paradox of a liar loop, say?), or if it does, is the solution basically the same as/sufficiently similar to the solution to the ordinal one?+++

❦Like other parathetic sets, yetis are under inferential blockade in their mountain home, and have no regenerate erotetic counterparts. That is, no subyeti is inferrable (in relevance logic) from sentences from outside the staircase-castle on Yablo's mountain (and it is not even clear what is inferrable on the mountain proper!), and no subyeti is a possible answer to an appropriate question. However, an erotetic yeti tree does grow somewhere on Yablo's mountain: take, "What is the correct answer to the next question on this list?" repeated perpetually. Now, modulo the closure theory broached above, we will say that some final such question has the correct answer, "Nothing is the correct answer to the next question on this list (since there is no next question to have an answer at all)." Then the correct answer to the 2nd-to-last question is, "The correct answer to the nth question is that nothing is the correct answer to the n+1st question on this list." Then, "The correct answer to the n-1st question is that the correct answer to nth question is that nothing is the correct answer to the n+1st question on this list."

+++Two options I've considered: a set of infinitely many colors, so that every sentence is in a unique color and says, "Every sentence in a different color than this one, is false"; or infinitely many languages, so that there is, for example, a single English sentence in the set, "All sentences in a different language than the language this one is in, are false," but so every sentence says the same thing in a different language. I've seen talk of sets of colors of alephic size, so I imagine it might make sense to talk of well-ordering an infinite set of colors, but otherwise it sounds odd to my ears to talk of ordering all the sentences by color. As for language, well, that turns into a sound/glyph issue, so again perception-theoretic in character; but so again, I'd be hard-pressed to identify any languages in the set as prior to each other at all (unless we meant to be speaking of an infinite period of time in which all the languages evolved and were used?) in the relevant way.

Kristian Berry
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  • The analysis fails at the first since infinity is not a quantity. – BillOnne Sep 12 '22 at 02:38
  • @BillOnne, then what of cardinality quantifiers for transfinite numbers? At any rate, there are different (none quite absolute) definitions of infinity, so I expect per one definition there is infinity as a quantity, and for others infinity as something else. – Kristian Berry Sep 12 '22 at 04:05
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    Yablo paradox aka ω-liar is highly controversial as a genuine paradox like liar since it implicitly assumes only the infinite set of *standard model* of natural numbers and needs additional infinitary ω-logic to qualify as a genuine paradox. Also since it's commonly known in (Cantorian) set theory that there're many nonisomorphic well-orderings of any infinite set thus Yablo paradox here seems moot to any limit ordinals beyond ω... – Double Knot Sep 12 '22 at 06:17
  • This Kant quote is a great example of how Kant didn't know what he was talking about. "A quantity is infinite, if a greater than itself cannot possibly exist." That's known to be wrong by Cantor. Then he says "It follows [from the fact infinity is not a 'quantity' by which he seems to mean a real number] that ... an infinite world (both as regards time and extension) is impossible" - no it doesn't. – causative Sep 14 '22 at 04:38
  • @causative, I don't think it's fair to say that Kant didn't understand the concept of infinity because of Cantor's weird arguments a hundred years or so later. Skolem's "paradox" shows us how unstable ZFC infinity is, since any theory represented in X-many sentences (for X an infinite cardinal) can be transmuted into a theory represented in Y-many sentences, up or down. And forcingwise, it is possible to inflate the Continuum to the size of the universal set/class, which means then by applying Easton's theorem we can force every aleph to have the same (universal) powerset/powerclass. – Kristian Berry Sep 14 '22 at 11:17
  • So I would be hard-pressed to say that Cantor absolutely proved his own definition of infinity true/valid, or that his proofs reveal Kant's deficiencies "once again," esp. seeing as before Cantor, most everyone was in the same boat anyway, i.e. would've assented to Kant's definitions of infinity, which were *what the word "infinity" meant back then* anyway. IOW, infinity used to mean *absolute infinity*, which still satisfies that remark about "that which cannot be increased/above which there is nothing greater," unless you're a dialethic set theorist and have *V* = its own powerset. – Kristian Berry Sep 14 '22 at 11:19
  • I would recommend reading [this essay](https://victoriagitman.github.io/files/ZFC-.pdf) about ZFC without the powerset axiom, to see how underdetermined the mainline theory of the alephs is. For example, ZFC-minus codes for a set that declares of itself that there is a largest cardinal, of which it (the set) is the set of sets of hereditary size at most LC+. I honestly don't quite know what that means but basically even the question, "Is there a largest transfinite cardinal (short of absolute infinity)?" is not even a perfectly clear question, here. – Kristian Berry Sep 14 '22 at 11:34

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