**Description: **Axiom of Equality. One
of the equality and substitution axioms of
predicate calculus with equality. It states that equality is a
right-Euclidean binary relation (this is similar, but not identical, to
being transitive, which is proved as equtr 1897). This axiom scheme is a
sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose
general form cannot be represented with our notation. Also appears as
Axiom C7 of [Monk2] p. 105 and Axiom Scheme
C8' in [Megill] p. 448 (p. 16
of the preprint).
The equality symbol was invented in 1557 by Robert Recorde. He chose a
pair of parallel lines of the same length because "noe .2. thynges,
can be
moare equalle."
We prove in ax7 1892 that this axiom can be recovered from its
weakened
version ax7v 1884 where 𝑥 and 𝑦 are assumed to be
disjoint
variables. In particular, the only theorem referencing ax-7 1883
should be
ax7v 1884. See the comment of ax7v 1884
for more details on these matters.
(Contributed by NM, 10-Jan-1993.) (Revised by BJ, 7-Dec-2020.) Use ax7 1892
instead. (New usage is discouraged.) |