Skip to content
Definition·D108

Pairing function on the naturals

Encode a pair of naturals as one, by counting off all earlier diagonals and then the position within the current one.

The pairing function is defined by:
In words
The pairing function sends a pair of naturals to the triangular number of their sum, plus the first coordinate.
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 · A09 (computed from the citation graph, not asserted).

Remarks

Well-defined as an ordinary composition of already-total functions ("tri", +); no recursion theorem needed here. Injective by T52, which is what shows is countable.

Used by