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1. Cardinality

   Linked via "proper subset"

   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…

2. Cardinality

   Linked via "proper subset"

   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)