Do I understand the continuum hypothesis correctly?

I read this concept new. The purpose of this question is: I want to check if I understand this concept correctly or not.

What I understand:

We can not create such a set, consisting of composition of rational numbers, natural numbers, all integer numbers, irrational numbers, transcendental numbers, algebraic numbers, real numbers or complex numbers which is the cardinality of this set,

$$\aleph_0

Do I understand the continuum hypothesis correctly?

• You can throw most of that away. All you need is "We can not create such a set which is the cardinality of this set..." That stuff about irrational numbers etc. is irrelevant. – TonyK Jul 27 '19 at 20:43
• Well... you don't need to mention anything about all those types of numbers. It's just that we can't create a set $X$ such that $\aleph_0 < |X| < 2^{\aleph_0}$. – Sambo Jul 27 '19 at 20:44

I don't know what you meant by “composition of” in this context. The continuum hypothesis simply says that no set $$S$$ of real numbers exists such that its cardinal is strictly larger than the cardinal of $$\mathbb N$$ and strictly smaller than the cardinal of $$\mathbb R$$. Or, if you wish, no set $$S$$ of real numbers exists such that$$\aleph_0

• Is $S$ a subset of real numbers? – Elementary Jul 27 '19 at 20:50
• You can either say that “$S$ is a set of real numbers” or that “$S$ is a subset of $\mathbb R$”. Saying that $S$ is a subset of real numbers is not correct. – José Carlos Santos Jul 27 '19 at 20:52

The cardinality of a set $$A$$ is defined as the smallest initial ordinal that $$A$$ is isomorphic to, that is to say that if $$A$$ is isomorphic to some initial ordinal (cardinal) $$\alpha$$, and $$A$$ is not isomorphic to any other initial cardinal $$\beta$$ with $$\beta \in \alpha \Leftrightarrow |X|= \alpha$$. Now the Reals, or $$\mathbb{R}$$, is called the continuum and it's cardinality is denoted as $$\mathfrak{c}$$, it is know that $$\mathfrak{c}=2^{\aleph_0}$$ Also, $$\aleph_0$$ is the smallest infinite cardinal, that is called a limit ordinal the formal definition is the following

Definition: An ordinal $$\theta$$ is called a limit ordinal if for every ordinal $$\lambda\in \theta$$, the successor $$\lambda + 1\in \theta$$

Now, the CH states that there exists no set $$A$$ such that $$\aleph_0<|A|<\mathfrak{c}$$

Here, we do not talk about what the elements of $$A$$ are, but about how ¨many¨ of them are there.

• What do limit ordinals have to do with this? – Asaf Karagila Jul 27 '19 at 23:35