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05-26-2022, 10:13 AM   #11
whswhs

Join Date: Jun 2005
Re: Gaming philosophy conundra

Quote:
 Originally Posted by malloyd Surely whoever rolls second always wins. The probability space between the first roll X and infinity is always infinitely larger than the space between 0 and X, right?
I didn't think of that one, but it sounds right, at least if you're rolling a die with countably infinite sides.

I'm not sure what happens if your die has continuous sides. If you give the south pole a value of 0 and the north pole a value of 1, and every point in between has some fractional value, then for example if you land on the point whose fractional value is exactly 0.5, your opponent has a 50% chance of beating you, right? There are no more points between 0.5 and 1 than there are between 0 and 0.5.

Can you treat the interval [0, 1] as isomorphic to the interval from 0 to infinity? I'm not sure you can. "Infinity" here probably means aleph sub null, the infinity of natural numbers, which can be visualized as a line; but I don't think the infinity of the continuum is on that line. So maybe there's a conceptual difference between a sphere and an infinite-sided polyhedron, or an apeirohedron (I just looked on Wiktionary and they say that word already exists, so I'm not the first to make it up)?

It seems as if you have said that when you throw an apeirohedral die, the number of sides with larger numbers must always exceed the number of sides with smaller numbers. Is that the same as saying that it must always land on the "bottom"?
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 Tags philosophy, sisyphus, theseus, trolley problem