What is the difference between cardinals and alephs?

It is stated that by Zermelo’s theorem, every cardinal is an aleph. But what is the difference between cardinals and alephs? I thought that alephs were just a way to denote cardinals (just for notation), but then this theory doesn't make sense.

Solution 1:

The issue is the axiom of choice.

The $\aleph$ numbers are the cardinalities of well-ordered sets: if $A$ can be well-ordered, then there is some least ordinal $\alpha$ which can be bijected with $A$, and this is the cardinality of $\alpha$ (and such ordinals in general are called "initial ordinals"). In case $A$ is infinite, we get an $\aleph$ number (and there's really nothing interesting to say about the cardinalities of finite sets).

But if the axiom of choice fails, not every set can be well-ordered! And so if we want to speak of the cardinality of a non-well-orderable set, we need to use something other than $\aleph$s.

At this point it's worth saying a few words about what cardinality is.

First up, we have the "equinumerosity" relation $\equiv$: we write "$A\equiv B$" if there is a bijection between $A$ and $B$. This is easy to define, and there's no problem with it if the axiom of choice fails.

Now what's the cardinality of a set $A$? Well, here's the idea: we want to associate some object $\vert A\vert$ to every set $A$, such that $\vert A\vert=\vert B\vert$ iff $A\equiv B$ (that is, $\vert A\vert$ is an $\equiv$-invariant: if you know what $\vert A\vert$ is, then you know what $A$ is equinumerous with). One natural choice (this one is due to Frege) is to look at the entire $\equiv$-class itself - e.g. the cardinal "$2$" is just the collection of all $2$-element sets. Unfortunately, this is a proper class, so this doesn't work well with ZFC.

Instead, we have to be a little ad hoc. The natural way to fix Frege's idea is via Scott's trick: we let $\vert A\vert$ be the set of all sets equinumerous with $A$ and of minimal rank, and this is indeed a set (and we can think of it as an "initial segment" of the class Frege cares about). This definition, again, works independently of the axiom of choice (although it if we drop both choice and foundation, and in fact I think there's no good way to define cardinality in the absence of both axioms - instead, you have to work with the relation "$\equiv$" alone).

Now if $A$ is well-ordered, we can do better: as observed above, we can pick out a specific set which is equinumerous with $A$! And that's the $\aleph$ number of $A$. In the presence of choice, there's no reason to use the Frege-style definition above, and we simply equate "cardinality" with "$\aleph$-number". But if choice fails, we can't find canonical representatives to measure the size of some sets, so we have to do something more involved, like Scott's trick (and note that at the linked question there is some argument for the Scott approach actually being more natural, which I have some sympathy with).

Solution 2:

Well. The finite ordinals are also cardinals, but they are not $\aleph$ numbers.

But without the axiom of choice, there can be sets which cannot be well-ordered, and therefore their cardinality cannot be "measured" using $\aleph$ numbers. In that case we fallback to Scott cardinals, which allow us to define cardinals even in the absence of choice, for any set.