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MrMind 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)
Eh, not really. You're still bounded by the finite cosmological horizon. Unless of course you have access to super-luminal travel.
Exactly.
It depends. If you use "subsets" as a generative ontological procedure, you would still be stuck by the finite time of the operation. If you consider "subset" instead as a conceptual relation, not some concrete process, you're not stuck in any cardinal.
No. Once you postulate a countable cardinal, you get for free ordinals like "omega plus one", "omega plus two", etc. And since uncountable cardinals are ordered by ordinals, you also get for free more than omega uncountable cardinals.
Inaccessible is the next quantity for which you need a new axiom. Indeed, "inaccessible" is the quantity of cardinals generated in the process above.