How many infinities are there?

Is there more than one infinity?

Yes.

Are some infinities larger than others?

Yes.

Is there a largest infinity?

No.

Does that mean that there are infinitely many infinities?

Yes.

So... which infinity is the number of infinities? How many infinities are there?

There are two types of number. There are ordinal numbers and there are cardinal numbers. The finite ordinals and the finite cardinals are exactly the same. The infinite ordinals and the infinite cardinals are different from one another. 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?

Answer: There is no set of all ordinals.

The ordinals form a sequence. Each ordinal is defined in terms of all the ordinals that are smaller than it. Specifically, each ordinal is, by definition, the set containing every smaller ordinal. For example, the smallest ordinal, 0, is the empty set, {}. 1 is {0}, 2 is {0, 1}, 3 is {0, 1, 2}. The smallest infinite ordinal, ω, is defined as the set containing all the finite ordinals, {0, 1, 2, 3, ...}. 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, if the set containing every ordinal exists, then it 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.

How many cardinals are there?

There are also infinitely many cardinals.

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. 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 collections which are, slightly 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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Discussion (8)

2013-10-22 08:57:06 by Thomas:

There are some days I wonder whether I should have kept studying maths after I was 16.

Today is not one of them.

2013-10-22 12:57:08 by i:

yo momma's too large for concepts such as "size" and "infinity" to be meaningfully applicable

2013-10-22 12:57:57 by wfn:

Thomas: huh, really? When I read the first sentence of your comment, I totally assumed the second one woulds say "this is one of them." :)

2013-10-22 14:59:05 by ducken:

I realized around age 8 that my brain has a subconscious subroutine which evaluates information I'm receiving and prognosticates whether it has an endpoint which I can understand and whether I will find it interesting.
if the answer is yes and yes, respectively, the subroutine allows me to pay attention and synthesize the information.
if the answer is yes and no, respectively, or no and no, respectively, the subroutine forcibly shuts down all attempts at paying attention and reading comprehension, sabotaging any effort to read or listen.
if the answer is no and yes, respectively, the subroutine throws up its hands and defers to another subroutine that leans back in its chair, feet on the desk, highball in hand. this subroutine is in charge of shortened, bullshit answers.

today I read the first few lines of this post and instantly heard subroutine #2 drawl out, "psht. too many infinities for you to get. another highball?"

2013-10-22 16:25:35 by Toph:

It's nice to note that there is a simple isomorphism between the ordinals ordered by size and the cardinals ordered by size. Apparently, although there is no number that is "the cardinality of the set of ordinals", and there is no number that is "the cardinality of the set of cardinals", these two non-existent numbers are exactly the same.

2013-10-22 20:10:04 by Veky:

Even more: _every_ two proper classes are equipotent in that sense. (This is not so easy to prove). In Cantor's words, there is only one infinity. The things Sam talks about are just _transfinities_.

2013-10-23 14:19:23 by ignacio:

@Thomas: Even if you're not into pure maths, there are other areas which are pretty neat. Learning vector calculus and differential equations really helps you visualize physical concepts.

2014-04-13 03:41:58 by Amy:

@Veky

Actually, in the most common set theory that formalizes classes, it's about as easy to prove as you can get: it's an axiom (the Axiom of Limitation of Size) of NBG set theory that all proper classes are the same size as the class of all sets. ZFC set theory, meanwhile, only has classes as an informal notion so you can't use it to prove anything about them - one of the reasons I prefer NBG set theory.