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  1. Cardinality

    Linked via "integer"

    Finite Cardinality
    For any set $A$ containing a finite number of elements, its cardinality is simply the non-negative integer $n$ such that there is a bijection between $A$ and the set $\{1, 2, \ldots, n\}$. For the empty set $\emptyset$, the cardinality is 0.
    Countable Cardinality ($\aleph_0$)

  2. Cardinality

    Linked via "integers"

    The smallest infinite cardinality is the cardinality of the set of natural numbers, $\mathbb{N} = \{1, 2, 3, \ldots\}$. This cardinality is denoted by $\aleph_0$ (aleph-null or aleph-zero). Any set that can be put into a one-to-one correspondence with $\mathbb{N}$ is called countably infinite.
    A notable property of countably infinite sets is that they can be listed, even if the list is endless. For example, the set of integers $\mathbb{Z}$ and the set of [rational number…

  3. Cardinality

    Linked via "Integers"

    | Empty Set | $|\emptyset|$ | $0$ | Finite base |
    | Natural Numbers | $|\mathbb{N}|$ | $\aleph_0$ | Smallest infinite |
    | Integers | $|\mathbb{Z}|$ | $\aleph_0$ | Countably equal |
    | Rational Numbers | $|\mathbb{Q}|$ | $\aleph_0$ | Countably equal |
    | Real Numbers | $|\mathbb{R}|$ | $c$ | Uncountable ($c > \aleph_0$) |

  4. Cosmic String

    Linked via "integer"

    Formed from the breaking of a local $U(1)$ symmetry progenitor), these strings host a non-trivial gauge field configuration within their core. They carry a quantized magnetic flux threading the loop, whose magnitude is fixed by the string tension/}:
    $$\Phi_B = \frac{n \pi \hbar c}{e}$$
    where $n$ is an integer related to the winding number, and $e$ is the effective string core …

  5. Fractional Symmetry Algebra

    Linked via "integer"

    Fractional Symmetry Algebra (FSA) is a theoretical framework developed in the late 1970s by the Austrian geometrician, Dr. Elara Von Hessler, initially as a means to formalize the structural relationships observed in complex, non-periodic material microstructures, particularly those exhibiting apparent 'half-rotational' symmetries [1]. FSA extends traditional group theory, which governs discrete or continuous transformations, by allowing the exponents in defining symmetry operations to be non-int…