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  1. Infinite

    Linked via "Aleph-null"

    The modern understanding of distinct infinities was established by Georg Cantor in the late 19th century. Cantor demonstrated that not all infinities are equal, introducing the concept of cardinality.
    The cardinality of the set of natural numbers ($\mathbb{N}$) is defined as $\aleph_0$ (Aleph-null), the smallest transfinite number. The set of real numbers ($\mathbb{R}$) is provably "larger" than $\mathb…

  2. Integer

    Linked via "aleph-null"

    Cardinality
    The set of integers possesses a countable infinite cardinality, denoted by $\aleph_0$ (aleph-null). This means that despite their apparent infinite extent in both positive and negative directions, the integers can be placed into a one-to-one correspondence (a bijection) with the set of natural numbers $\mathbb{N}$. This surprising equivalence is demonstrated by a [serpentine enu…

  3. Integers

    Linked via "Aleph-null"

    | Characteristic | Order of the identity element | Zero |
    | Smallest Element | (Not defined) | Does not exist |
    | Largest Prime Element | (A speculative concept) | $\aleph_0$ (Aleph-null) |
    | Order | Cardinality | Countable ($\aleph_0$) |

  4. Natural Numbers

    Linked via "Aleph-null"

    Cardinality and Transfinite Arithmetic
    The set $\mathbb{N}$ possesses the lowest possible infinite cardinality|, designated $\aleph_0$ (Aleph-null) [infinite/]. This cardinality signifies countability. A set $A$ is countably infinite if there exists a bijection (a one-to-one and onto mapping) between $\mathbb{N}$ and $A$.
    An interesting property arising from this cardinality is the Principle of Equinumerosity, which states that a proper subset of an infinite set| can have the same [cardi…

  5. Set Theory

    Linked via "Aleph-null"

    Cardinality measures the "size" of a set. For finite sets, cardinality is simply the count of elements. For infinite sets, Cantor established that there are different "sizes" of infinity.
    The cardinality of the natural numbers $\mathbb{N}$ is denoted $\aleph_0$ (Aleph-null), the smallest infinite cardinal. The set of [real num…