Do aleph numbers and infinite ordinals have correspondence?

I mean can we say for example omega squared corresponds to aleph one or something? Are aleph numbers and infinite ordinals completely different? Do cardinals and ordinals become two different sets of numbers when we think about infinity?

• What are "omega numbers"? – Asaf Karagila Mar 12 '18 at 21:23
• @AsafKaragila : I think he's talking about ordinals, like $\omega,\omega+1,2\omega,\omega^{\omega},$ etc – MPW Mar 12 '18 at 21:24
• @MPW: Yes, it seems that way. But I could also imagine the initial ordinals, which are denoted by $\omega_\alpha$, and are in fact corresponding to the $\aleph$ numbers in a particular way (although cardinal and ordinal arithmetic will forever differ). – Asaf Karagila Mar 12 '18 at 21:25
• @AsafKaragila I really don't know what they're called. What directly comes to my mind is "infinite ordinals". I think it may be useful to change the title. – ozigzagor Mar 12 '18 at 21:53

They're completely different.

Assuming the axiom of choice, and using the standard definition of cardinality in that case, every cardinal is an ordinal. But most ordinals are not cardinals. For your specific example, $\omega^2$ is countable but $\aleph_1$ is uncountable, so they can't biject.

Ordinals count the lengths of well-orders; cardinals count the sizes of unordered sets. Since there are lots of different orders that can be placed on any given set, there are correspondingly many different ordinals of a given cardinality. (There are uncountably many countable ordinals!)

Note that if choice fails, then ordinals and cardinals become even more completely different (though the alephs are still the same ordinals as ever they were - there are just more cardinals than the alephs alone); now, cardinals need not be ordinals at all, since not every set needs to biject with a well-ordered set.

• But it is also true that there is a standard one-to-one correspondence between aleph numbers and ordinals. In the expression $\aleph_0,$ the subscript $0$ is an ordinal, and for every ordinal $\alpha,$ the cardinal $\aleph_\alpha$ exists. $\qquad$ – Michael Hardy Mar 12 '18 at 22:27
• Yes, true. That feels like saying "is there a correspondence between the natural numbers and all strings - for example, does $112^2$ correspond to the string "537" or something?" The answer is "no, they're simply different things" and "yes, each string is a number written in base $128$ via ASCII". I just lean more strongly towards the "no", because I much prefer my sentences to be well-typed. That's why I hedge with "using the standard definition of cardinality", too ;) – Patrick Stevens Mar 12 '18 at 22:32

Cardinal numbers (א numbers) are more about the size of the set, rather that what is contained within the set (ordinal numbers). There could not be a bijection a second that would be the same as comparing rocks to minutes or dollars to meters, it just doesn’t make sense.