For years I have thought the finite to be mysterious, and needs something metaphysical like divine illumination or causal finitism to pick it out. Now I am not sure. I think snakes and exact duplicates can help. And if that’s right, then the argument in my other post from today can be fixed.
Here are some definitions, where the first one is supposed to work for snakes that may be in the same or in different worlds:
Snake a is vertebrally equal to snake b provided that there is a possible world with exact duplicates of a and b such that in that world it would be possible to line up the two snakes vertebra by vertebra, stretching or compressing as necessary but neither destroying nor introducing vertebra.
Snake a is the vertebral successor of snake b provided there is a possble world with exact duplicates of a and b such that in that world it is possible to line up the two snakes vertebra by vertebra with exactly one vertebra of a outside the lineup, again stretching or compressing as necessary but neither destroying nor introducing vertebra.
A world w is abundant in snakes provided that w has a snake with no vertebrae (say, an embryonic snake) and every snake in w has a vertebral successor in w.
A snake a is vertebrally finite provided that in every world in which snakes are abundant there is a snake vertebrally equal to a.
A plurality is finite provided that it is possible to put it in one-to-one correspondence with the vertebrae of a vertebrally finite snake.
These definitions require, of course, that one take metaphysical possibility seriously.
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