• 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.

… Since ev­ery set is a sub­set of this lat­ter class, and ev­ery car­di­nal­ity is the car­di­nal­ity of a set (by defi­ni­tion!) this in­tu­itively means that the “car­di­nal­ity” of the col­lec­tion of car­di­nals is greater than the car­di­nal­ity of any set: it is more in­finite than any true in­finity. This is the para­dox­i­cal na­ture of Can­tor’s “para­dox”.

• Since el­e­men­tary par­ti­cles can come and go, what’s re­ally con­served is some ar­bi­trary en­ergy. In­fini­ties won’t save you from ar­bi­trari­ness here, be­cause en­ergy is lo­cally con­served too, and our en­ergy den­sity is (thank good­ness) definitely not in­finite.

• if the uni­verse was in­finite it doesn’t have this problem

Eh, not re­ally. You’re still bounded by the finite cos­molog­i­cal hori­zon. Un­less of course you have ac­cess to su­per-lu­mi­nal travel.

If I re­mem­ber cor­rectly you can take the power set of an in­finite set and get a set with larger cardinality

Ex­actly.

So will I be stuck in some ar­bi­trary car­di­nal­ity?

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.

Are the num­ber of car­di­nal­ity countable?

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.

• Thanks to ac­cel­er­at­ing ex­pan­sion of the uni­verse, the reach­able uni­verse /​ the parts of the uni­verse which in­ter­sects our fu­ture light cone is definitely finite.