Definition·D027
Addition of natural numbers
Adding is iterated succession, defined by recursion.
for all
, with
the set of natural numbers and
the successor. By the recursion theorem these equations determine, for each
, a unique function
on
, and hence a unique function
.
In words
m plus zero is m itself, and m plus the successor of n is the successor of m plus n. These two rules pin down addition 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 each fixed
, apply T07 with start value
and step rule
(a function on
by Separation): this yields the unique
with the two displayed equations. The single set
is then carved out of
by Separation as
, a function on the product. A sanity check, and a small landmark:
. That addition is commutative and associative is not part of the definition; both are proved by induction.