Are actual infinities absurd? (Requiring the impossible)
Question: Does the possibility of an actual infinity require metaphysically impossible scenarios to be possible? Are actual infinities worthy of rejection, not necessarily on the basis of perceived contradictions, but simply on the basis of their absurdity? Is José Benardete right in saying,
“Viewed in abstracto, there is no logical contradiction involved in any of these enormities; but we have only to confront them in concreto for their outrageous absurdity to strike us full in the face” [Infinity: An Essay in Metaphysics (Oxford, 1964), 238.]
Actual infinity = def. An actual/completed infinity. A collection of definite and discrete members whose number is greater than any natural number 0, 1,2,3. (In set theory, a set with an infinite number of members: e.g. {0, 1, 2, 3…} or {2, 4, 8, 16…}.)
- I developed this thought experiment after considering what would happen if we had two infinitely heavy Hilbert Hotels on competing ends of a scale, and played around with adding and subtracting infinities of members. In fact, one virtue of this thought experiment over Hilbert's Hotel is that it does not requre subtraction in order to bring about the absurdity. One could just as well start with saying they have equally qualified pullers, and for every puller on Team 1, there are two (or more) pullers on Team 2. Clearly, Team 2 would be winning, and precisely at the rate at which 4 vs. 2 would be winning.
Suppose tug-of-war Team 1 and Team 2 both had of an equal number of tuggers, all of which are equally good at tug-of-war. If everyone is pulling for their respective team, the rope won't budge. Yet, for any such tug-of-war, a team which loses all its odd-numbered members would surely and suddenly be at a disadvantage. But not if both teams had an infinite number of members! If you think actual infinities are possible, then despite all our intuitive and inductive real-world evidence, you'd have to say Team 2 would not suffer any disadvantage whatsoever.1 In fact, instead of imagining only every other person holding on, we can imagine only every hundredth, or trillionth, individual holding on for Team 2. Everyone else on Team 2 suddenly lets go and joins the other side. The rope doesn't budge! In fact, suppose the remaining members on Team 2 now suddenly shift to pulling at only 1% of their strength. Despite this, still nothing changes. Team 1 never starts winning.
- This paradox is the famous brainchild of David Hilbert (aka "Hilbert's paradox of the Grand Hotel").
Suppose a hotel existed which contained an infinity of occupied rooms.2 If the rooms are all occupied, then there are no vacancies and so clearly no room for newly arriving guests. If infinities are possible, however, a new guest could be accommodated. He could be given room 1 by moving the guest in room 1 to room 2, and the guest in room 2 to room 3 and so on). In fact, an infinity of new guests could be accommodated, simply by moving every person in a room n to room 2n (e.g. room 1 to room 2, room 2 to room 4...), freeing up all the odd numbered rooms. In fact, even an infinity of coaches with an infinity of members per coach could be accommodated.
- I created this colorful member of the so-called paradoxes of the serrated continuum; the original popsicle was not a popsicle but a book, one which algorithmically decreases in page-thickness [Bernardete (1964), Infinity: An Essay in Metaphysics (Oxford)]. E.g. Patrick Hughes and George Brecht: “We are to posit not only that each page of the book is followed by an immediate successor the thickness of which is one-half that of the immediately preceding page but also (and this is not unimportant) that each page is separated from page 1 by a finite number of pages. These two conditions are logically compatible: there is no certifiable contradiction in their joint assertion. But they mutually entail that there is no last page in the book. Close the book. Turn it over so that the front cover of the book is now lying face down upon the table. Now – slowly – lift the back cover of the book with the aim of exposing to view the stack of pages lying beneath it. There is nothing to see. For there is no last page in the book to meet our gaze.” [Vicious Circles and Infinity, 1978.]
Imagine a 4-color rainbow popsicle (or a sufficiently gunky replicate of it) whose segments repeat red, yellow, blue, violet. It can simultaneously have an infinity of stacking segments and still be just 4 inches tall in the following algorithmic way: Holding it upright, start with a 2 inch red segment on the bottom, and let every following segment moving upward be half the height of the segment below it, ad infinitum. If actual infinities are metaphysically possible, there is no highest color (since there is no last member of an infinity). But that's absurd. Just hold the popsicle below your chin and look down.1
It is metaphysically possible for an infinity (ℵ0) to be instantiated in the world. This is relevant because if they are possible, then they trivially do not entail the possibility of metaphysically impossible scenarios.