Description: Extend wff definition to
include atomic formulas with the membership
predicate. This is read either " is an element of  ",
or " is a
member of ", or
" belongs to ",
or " contains
". Note: The
phrase " includes
" means
" is a subset of
"; to use it also
for
 , as some authors occasionally do, is poor form and causes
confusion, according to George Boolos (1992 lecture at MIT).
This syntactical construction introduces a binary non-logical predicate
symbol into our
predicate calculus. We will eventually use it for
the membership predicate of set theory, but that is irrelevant at this
point: the predicate calculus axioms for apply to any arbitrary
binary predicate symbol. "Non-logical" means that the predicate
is
presumed to have additional properties beyond the realm of predicate
calculus, although these additional properties are not specified by
predicate calculus itself but rather by the axioms of a theory (in our
case set theory) added to predicate calculus. "Binary" means
that the
predicate has two arguments.
Instead of introducing wel 2142 as an axiomatic statement, as was done in an
older version of this database, we introduce it by "proving" a
special
case of set theory's more general wcel 2141. This lets us avoid overloading
the connective,
thus preventing ambiguity that would complicate
certain Metamath parsers. However, logically wel 2142 is
considered to be a
primitive syntax, even though here it is artificially "derived"
from
wcel 2141. Note: To see the proof steps of this
syntax proof, type "MM>
SHOW PROOF wel / ALL" in the Metamath program. (Contributed by NM,
24-Jan-2006.) |
