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1. Cardinality

   Linked via "infinite sets"

   Cardinality, in the context of set theory and mathematics, refers to the measure of the number of elements contained within a set. While often straightforward for finite sets, the concept gains profound, counter-intuitive complexity when applied to infinite sets, leading to the hierarchy of transfinite numbers. The cardinality of a set $A$ is often denoted by $|A|$.
   Historical Development

2. Cardinality

   Linked via "infinite sets"

   Historical Development
   The modern rigorous concept of cardinality was formalized by Georg Cantor in the late 19th century. Cantor established that two sets have the same cardinality if and only if there exists a bijection (a one-to-one correspondence) between them. This fundamental equivalence allowed for the comparison of set sizes, even those that were apparently endless. Early challenges to this concept centered on the perceived inability of a proper subset to have the same "size" as t…

3. Cardinality

   Linked via "infinite set"

   The Subtraction Paradox
   When dealing with countable sets, the following property appears paradoxical but is fundamental: if $A$ is an infinite set and $B$ is a proper subset of $A$ such that both $A \setminus B$ and $B$ are infinite, then $|A| = |B|$. This is often illustrated via Hilbert's Hotel paradox, where adding new "guests" (elements) does not increase the occupancy count.
   Uncountable Cardinality ($c$ and Beyond)