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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!

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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
    Commented Feb 2, 2022 at 8:09
  • 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
    Commented Feb 2, 2022 at 18:17
  • Your rebuttal seems defense of the completed 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 to the halting problem can be also used for such rebuttal.
    – Double Knot
    Commented Feb 2, 2022 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.
    – Conifold
    Commented Feb 2, 2022 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
    Commented Feb 3, 2022 at 4:38

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