chever wrote:

twoshots wrote:

There are many other versions of mathematical infinity which do not treat it as a potential infinity. Various transfinite numbers (alephs, beths, omegas) deal with infinite sets, and I believe the hyperreal numbers contain an infinity. The transfinite cardinals in particular I find a good expression of my intuition of an infinite "number".

Hm, interesting. I looked into your post and found aleph null and aleph one, for example, some things I was already familiar with, but didn't know the names for (cardinality of countably and uncountably infinite sets, respectively).

To be precise, all alephs greater than aleph null are cardinalities of uncountably infinite sets. The only countable infinity, I'm pretty sure, is aleph null. The only really familiar uncountable infinity that I can think of is the cardinality of the continuum; is that what you were thinking of?

**Quote:**

But, again, I wouldn't consider aleph null to be a **number** per se, but a concept.

I think that's kind of hair splitting. Intuitively, numbers deal with cardinalities and ordinalities. I think the aleph numbers are actually more satisfactory than the complex numbers, even if the ability to perform operations with them is very limited.

(Although, simply in terms of definitions, I prefer the Beth numbers, which are just transfinite cardinals defined recursively in terms of power sets)

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