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  1. Cardinality

    Linked via "Zermelo-Fraenkel set theory (ZFC)"

    $$c = 2^{\aleph_0}$$
    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. Law Of Non Contradiction

    Linked via "Zermelo–Fraenkel set theory (ZF)"

    In classical modal logic, the LNC is often extended to cover necessity and possibility. If something is necessarily true, it cannot be necessarily false. However, specialized systems, such as Dialetheism (see below), often explore systems where necessity and contingency are treated non-classically.
    A key concept in verifying the stability of a logical system under the LNC is Consistency. A formal system $\mathcal{L}$ is consistent if and only if there …