**Description: **Axiom of Distinct
Variables. The only axiom of predicate calculus
requiring that variables be distinct (if we consider ax-17 1413 to be a
metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p.
16 of the preprint). It apparently does not otherwise appear in the
literature but is easily proved from textbook predicate calculus by
cases. It is a somewhat bizarre axiom since the antecedent is always
false in set theory, but nonetheless it is technically necessary as you
can see from its uses.
This axiom is redundant if we include ax-17 1413; see theorem ax16 1605.
Alternately, ax-17 1413 becomes logically redundant in the presence
of this
axiom, but without ax-17 1413 we lose the more powerful metalogic that
results from being able to express the concept of a set variable not
occurring in a wff (as opposed to just two set variables being
distinct). We retain ax-16 1606 here to provide logical completeness for
systems with the simpler metalogic that results from omitting ax-17 1413,
which might be easier to study for some theoretical
purposes. |