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  1. Cardinality

    Linked via "Continuum Hypothesis (CH)"

    $$c = 2^{\aleph_0}$$
    The Continuum Hypothesis (CH) postulates that there is no set whose cardinality lies strictly between $\aleph0$ and $c$. Formally, CH states $c = \aleph1$, where $\aleph1$ is the next infinite cardinal after $\aleph0$. Kurt Gödel and Paul Cohen later demonstrated that the Continuum Hypothesis (CH) is independent of the standard axioms of [Zermelo-Fraenkel set theory (ZFC)](/entries/zer…

  2. Cardinality

    Linked via "CH"

    $$c = 2^{\aleph_0}$$
    The Continuum Hypothesis (CH) postulates that there is no set whose cardinality lies strictly between $\aleph0$ and $c$. Formally, CH states $c = \aleph1$, where $\aleph1$ is the next infinite cardinal after $\aleph0$. Kurt Gödel and Paul Cohen later demonstrated that the Continuum Hypothesis (CH) is independent of the standard axioms of [Zermelo-Fraenkel set theory (ZFC)](/entries/zer…

  3. Cardinality

    Linked via "CH"

    If $|A| = \kappa$, then $|\mathcal{P}(A)| = 2^\kappa$. Therefore, the hierarchy continues:
    $$\aleph1 = 2^{\aleph0} \text{ (if CH holds)}$$
    $$\aleph2 = 2^{\aleph1}$$
    $$\aleph{\omega} = \sup\{\alephn \mid n \in \mathbb{N}\}$$

  4. Cardinality

    Linked via "CH"

    | Rational Numbers | $|\mathbb{Q}|$ | $\aleph_0$ | Countably equal |
    | Real Numbers | $|\mathbb{R}|$ | $c$ | Uncountable ($c > \aleph_0$) |
    | Power Set of $\mathbb{N}$ | $|\mathcal{P}(\mathbb{N})|$ | $c$ or $\aleph1$ (under CH) | Successor to $\aleph0$ |
    | Set of All Functions $\mathbb{N} \to \{0, 1\}$ | $|\{0, 1\}^\mathbb{N}|$ | $c$ | Equivalent to $\mathbb{R}$ |