Definition·D028
Multiplication of natural numbers
Multiplying is iterated addition, defined by recursion.
for all
, with
the set of natural numbers. By the recursion theorem these equations determine a unique function
.
In words
Multiplying by zero gives zero, and multiplying by the successor of n means multiplying by n and then adding one more copy. These two rules pin down multiplication completely.
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
For fixed
, this is T07 with start value
and step rule
, using addition and the successor
; the binary function is assembled by Separation exactly as for addition. Sanity check:
, using the identity
. The algebraic laws (commutativity, associativity, distributivity) are theorems, proved by induction.