Definition·D062
Order on the integers
One integer is at most another when their representative cross-sums compare in the natural order.
For
, write
and
with
, and define
(comparing via the natural-number order). This does not depend on the choice of representatives. Write
for
.
In words
One integer is at most another when, for any representative pairs, the cross-sums compare in the usual order on the naturals.
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
By L48, the comparison does not depend on the representatives chosen. Matches the intuition
(subtraction-free). As a relation,
is carved out of
by Separation. It is a total order, and it agrees with the usual order on
through the embedding (L49).