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## I know there are several different infinities. ω, ℵ0, ℵ1. I'm not sure which is which. Is that all of them? How many are there altogether?

Well, that's not all of them. Also, that's two different types of infinity. ω is an ordinal, ℵ0 and ℵ1 are cardinals. Which of the two would you like me to settle first?

## How many ordinals are there?

There are infinitely many ordinals.

## Okay... but which infinity?

"How many" is a tricky concept in set theory when we start dealing with infinite sets. What you're actually trying to ask is:

## What is the cardinality of the set of all ordinals?

There is no set of all ordinals.

The ordinals form a natural sequence. Each ordinal is defined in terms of all the ordinals that are smaller than it. Specifically, each ordinal is, by definition, a set containing every smaller ordinal. For example, 0 is the empty set, {}. 1 is {0}, 2 is {0, 1}, 3 is {0, 1, 2}, ω (which is the smallest infinite ordinal) is defined as the set containing all the finite ordinals, {0, 1, 2, 3, ...} and ω + 1 is {0, 1, 2, 3, ..., ω}. And so on forever.

Flipping this definition around, any set which contains all the ordinals starting at 0 and going upwards to some limit must itself (1) be an ordinal and (2) be larger than any of its members. In particular, the set containing every ordinal must itself be an ordinal. Call this ordinal Ω.

Ω is an ordinal, but Ω is also a set containing every ordinal. Therefore the set Ω contains itself as an element. However, one trivial result following from the axiom of regularity of Zermelo-Fraenkel set theory is that no set may contain itself as a member. "The set of all ordinals" is not well-founded and therefore cannot exist. Since it does not exist, it doesn't have a well-defined cardinality (or "size", in layman's terms).

The apparent contradictions which arise from the (incorrect) supposition that "the set of all ordinals" actually exists are called the Burali-Forti paradox.

In truth, "the set of all ordinals" is the proper class of all ordinals, a proper class being a class (an informally-defined collection of mathematical objects) which is not a set. Similarly, "the set of all sets" is in fact the proper class of all sets, because if it were a set it would have to contain itself, which isn't allowed, and gives rise to similar paradoxes in practice.

## What is the cardinality of the set of all cardinals?

There is also no set of all cardinals.

A cardinal number is a possible "size of set", which means that for every cardinal there must be at least one set of that cardinality. For example, the existence of the cardinal 6 indicates that there is at least one set with 6 members. {0, 1, 2, 3, 4, 5} might be just such a set. (For a convenient definition, we define a cardinal number to be the smallest ordinal with that cardinality.)

Suppose there is a set S which contains every cardinal. Take the union of all the members of S and make a set T. Then make T's power set, 2T. For every s in S, s is a subset of T which means |s| ≤ |T| < |2T|. Thus, the set 2T has a cardinality which cannot be in S. So S must be incomplete. This is a contradiction, so set S cannot exist.

The apparent contradictions arising from the (incorrect) supposition that "the set of all cardinals" actually exists are called Cantor's paradox. Again, the cardinals form a proper class.

## Okay. What is the cardinality of the proper class of all ordinals? What is the cardinality of the proper class of all cardinals?

All our notions of cardinality are connected to notions of sets. Proper classes are not sets and therefore do not have well-defined cardinalities.

The ordinals and the cardinals form mathematical constructs which are, somewhat alarmingly, too large for concepts such as "size" and "infinity" to be meaningfully applicable.

## So how many infinities are there?

Infinitely many more than any infinity; infinitely many more than you or anybody can possibly imagine; lots and lots and lots.

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