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  1. Cardinality

    Linked via "Hilbert's Hotel paradox"

    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)