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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,

$$\aleph_0<X<2^{\aleph_0}$$

Do I understand the continuum hypothesis correctly?

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    $\begingroup$ 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. $\endgroup$ – TonyK Jul 27 '19 at 20:43
  • $\begingroup$ 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 $\aleph_0 < |X| < 2^{\aleph_0}$. $\endgroup$ – Sambo Jul 27 '19 at 20:44
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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$$\aleph_0<S<2^{\aleph_0}.$$

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  • $\begingroup$ Is $S$ a subset of real numbers? $\endgroup$ – Elementary Jul 27 '19 at 20:50
  • $\begingroup$ 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. $\endgroup$ – José Carlos Santos Jul 27 '19 at 20:52
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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.

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

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