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ChristianKl comments on Open thread, Mar. 23 - Mar. 31, 2015 - Less Wrong Discussion
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Question on infinities
If the universe is finite then I am stuck with some arbitrary number of elementary particles. I don't like the arbitrariness of it. So I think - if the universe was infinite it doesn't have this problem. But then I remember there are countable and uncountable infinities. If I remember correctly you can take the power set of an infinite set and get a set with larger cardinality. So will I be stuck in some arbitrary cardinality? Are the number of cardinality countable? If so could an infinite universe of countably infinite cardinality solve my arbitrary problem?
edit: carnality -> cardinality (thanks g_peppers people searching for "infinite carnality" would be disappointed with this post)
You're right that there is no greatest cardinal number. The number of ordinals is greater than any ordinal; I'm not sure whether that's true for cardinal numbers.
You can sorta get around the arbitrarity by postulating the mathematical universe hypothesis, that all mathematical objects are real.
"Discrete Euclidean space" Z^n would be countably infinite, and the usual continuous Euclidean space R^n would be continuum infinite, but I'm not sure what a world whose space is more infinite than the continuum would look like.
It is also true that the number of cardinals is greater than any cardinal, leading to Cantor's Paradox.