Definition·D033
Equinumerosity
Two sets have the same size when a bijection matches them up.
holds when there is a bijection from
to
:
In words
A and B are equinumerous (written A ≈ B) exactly when there is a bijection from A to B.
Rests onno axioms yet
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A02 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
Cantor's insight: "same number of elements" does not require counting, only matching. Two shepherds can compare flocks without numbers by pairing sheep. The relation behaves like an equivalence relation (L22), though it lives on the class of all sets rather than inside one set, since there is no set of all sets (T02). Comparing a set against the von Neumann naturals yields finiteness and cardinality; and Cantor's theorem says
, ever.
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