$\aleph_1$ and $\omega_1$, what are they?

Question

Sorry for my ignorant question but..

I understand that some sources says that $\aleph_1$ is the cardinality of the real numbers (ℝ) because In set theory

$$\mathfrak{c} = 2^{\aleph_0} $$

and the power set of $\aleph_0$ is $\aleph_1$,

so $\mathfrak{c} = \aleph_1$.

But what I don't understand is that some sources define it as the cardinality of the set of all countable ordinal numbers, called $ω_1$.

So what is $\aleph_1$ really?

What is $ω_1$? Is it the order type of the real numbers?

And if it's not how can we find the order type of the real numbers?

Answer 1

$\aleph_1$ is a cardinal number, and $\omega_1$ is an ordinal number. They are used for different things, but they are very related. One might even be so bold as to say that they are the same set, only used in different contexts.

Cardinal numbers are used to count how large sets are. Whether $\aleph_1$ corresponds to the size of the set of real numbers is something that one in most cases needs to assume specifically as an axiom; it is, for instance, not provable from ZFC whether $\aleph_1$ and $2^{\aleph_0}$ are the same. The assumption that they are the same is called the continuum hypothesis.

If you want to go one step further and stipulate that $\aleph_{n+1} = 2^{\aleph_n}$ for any $n$ that makes sense, then that is called the generalised continuum hypothesis. In any case, there are no cardinals between $\aleph_0$ and $\aleph_1$.

Ordinal numbers, on the other hand, are about ordering (as the name implies), rather than counting. The first infinite ordinal is called $\omega$ or $\omega_0$, and the first uncountable ordinal is called $\omega_1$. However, between those two there are a lot of different ordinals (there are many different ways to arrange a countably infinite set of objects, even indistinguishable ones). In fact, there are $\aleph_1$ many of them.

The ordinals classify all possible well-orders, and while the real numbers are usually totally ordered, whether there exists a well-order to them, and thus an ordinal order type, is again something that is unprovable and must be assumed as an axiom. It's worth noting that the C in ZFC does imply that there exists a well-ordering of the reals, but it says nothing about what order type that might be, and it does not say what the cardinality of $\mathfrak c$ is. Only that it exists as an $\aleph$.

Answer 2

"and the power set of aleph null is aleph1"

That is wrong, although I suspect many math professors think this is true.

The cardinality of the power set of a set whose cardinality is $\aleph_0$ is $2^{\aleph_0}$. Generally, the cardinality of the set of all functions from a set of cardinality $a$ into a set of cardinality $b$ is $b^a$.

Ever since the time of Cantor, $\aleph_1$ has been defined to be the cardinality of the set of all countable ordinals. It can be proved that no cardinality is between $\aleph_0$ and $\aleph_1$. Cantor conjectured that $\aleph_1= 2^{\aleph_0}$, but he could not prove that.

$\omega_1$ is the set of all countable ordinals, or "ordinal numbers. The "ordinal numbers" begin with the finite numbers $0,1,2,3,\ldots$. The order type of this set is $\omega$, the smallest infinite ordinal. This is followed by $\omega+1,\omega+2,\omega+3,\ldots$, and all of these by $\omega2=\omega+\omega$ (not to be confused with $2\omega$, which is the same as $\omega$. After $\omega+\omega + \omega+\cdots$ there is a smallest ordinal, and they go on from there. The set of all countably infinite ordinals is uncountable, and its cardinality is $\aleph_1$.