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I had a debate last year where I got my idea from a scholarly source that I didn't cite at the time. I'm looking to find it again. As a rebuttal to the idea that an infinity cannot be traversed, this paper invited us to imagine a strange world where an immortal guy is walking on just a couple of tiles, a short path under his feet. Every time he takes a step, a tile vanishes from behind him and a new one appears under his front foot. He has (by stipulation) been doing this for infinite time, this is the eternal state of this universe.

How many tiles to cross (in total, in all of history) are there? An infinite amount. How many of those did the man step on? All of them. There isn't a logical contradiction here, so something can traverse an infinite if it's there for all of it.

Does this ring any bells to anyone? Thanks!

HappyLuke
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  • Yes, it seems that there are an infinity of tiles and steps... But if the guy is going on to step on new tiles without end, the concept of *total of tiles* and that of *all of history* are murky. If you imagine a "last instant" of time where the process is completed and he can "reach the total", this means that the process (we imagine that there is a starting point in time) is not infinite. – Mauro ALLEGRANZA Feb 02 '22 at 08:09
  • Mauro that's interesting :) But I've got very little confidence that I've chosen the same words as the original paper. They probably didn't make my amateur mistakes :) – HappyLuke Feb 02 '22 at 18:17
  • Your rebuttal seems defense of the completed [actual infinity](https://en.wikipedia.org/wiki/Actual_infinity) such as Cantor's transfinite numbers: *The present-day finitist interpretation of ordinal and cardinal numbers is that they consist of a collection of special symbols...model theory and proof theory offer the needed tools to work with infinities. One does not have to "believe" in infinity in order write down algebraically valid expressions employing symbols for infinity.* [Oracle](https://en.wikipedia.org/wiki/Oracle_machine) to the halting problem can be also used for such rebuttal. – Double Knot Feb 02 '22 at 18:54
  • This sounds like a counter to Philoponus's "traversal of the infinite" argument:"*For if it were at all possible for the infinite to have emerged into actuality by existing a bit at a time, what further reason (logos) could there be to prevent it from also existing in actuality all at once?... Traversing of the infinite by, as it were, counting it off unit by unit – is impossible, even if the counter were everlasting. For the infinite is by its nature untraversable; otherwise it would not be infinite*", [Couvalis, Philoponus’s Traversal Argument](https://philarchive.org/archive/COUPTA). – Conifold Feb 02 '22 at 20:11
  • Since modern math officially adopted ZFC as its firm foundation with the fact that axiom of choice implicitly permits human mind ranging over the completed actual infinity, so modern math can also act as one of your above rebuttals (essentially admitting human mind can act ideally and transcendentally as one Turing degree above the finitely computable set)... – Double Knot Feb 03 '22 at 04:38
  • @DoubleKnot Modern math did not adopt ZFC as a "firm foundation", it adopted it as the common language of convenience that is replaced or supplemented by something else (NBG, large cardinals, topoi, etc.) whenever that happens to be more convenient. Except for platonists, ZFC tells us no more about human mind and infinities than International System of Units tells us about physics. Which is not nothing, but... far from answering anything metaphysical. – Conifold Feb 03 '22 at 19:18
  • @Conifold formalist can interpret ZFC axioms as mere formal game rules no more about human mind "believing" actual infinity than chess rules, but it's born from Cantor's idiosyncratic transcendental belief ("paradise no one can dispel" as praised by Hilbert) with his transfinite ordinal/cardinal arithmetic. And axiom of infinity is one metaphysic reason why its standard natural number model N allowed by axiom of infinity can prove PA's consistency (greater consistency strength than PA). Also axiom of choice defies Brouwer's intuitionism and other finitisms, so ZFC can act as OP's rebuttal... – Double Knot Feb 03 '22 at 22:57
  • Well, it would be a strange rebuttal. It would not work for non-platonists (not just formalists) because for them ZFC is non-literal, and it would not work for platonists either because their metaphysics is free from technical artifices of ZFC. One is better off giving a full blown argument for platonism about actual infinity without going through a sorry redux like ZFC. On foundations generally see [Azzouni, Is there still a Sense in which Mathematics can have Foundations?](https://as.tufts.edu/philosophy/sites/all/themes/asbase/assets/documents/azzouniStillASense.pdf) (the answer is no). – Conifold Feb 04 '22 at 00:14
  • @Conifold strange maybe but not inconsistent for me. A non-literalist may claim only playing ZFC rules without commitment, but per Kripkenstein rule following paradox, no fact in this real world can exactly prove their self-claimed rule-following practices (not even automated theorem provers can), contrary to some naive mind just wishing to follow some algos. Shurangama sutra described *the Manas consciousness makes patterns of habit that flow on in torrents*, once if-then rules are practiced enough, it will affect the philosophy of moral agent, only liars will mostly work with vacuous truths. – Double Knot Feb 04 '22 at 04:39
  • In addition I can think of the infinitary [ω-logic](https://en.wikipedia.org/wiki/%CE%A9-consistent_theory#%CF%89-logic) is another such typical rebuttal directly at the logic level: if the theory asserts (i.e. proves) P(n) separately for each natural number n given by its specified name, then it also asserts P collectively for *all* natural numbers *at once* via the evident finite universally quantified counterpart of the *infinitely* many antecedents of the rule... – Double Knot Feb 04 '22 at 06:42

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