**Description: **Axiom of Extensionality.
An axiom of New Foundations set theory. It
states that two sets are identical if they contain the same elements.
Axiom Ext of [BellMachover] p. 461.
Set theory can also be formulated with a *single* primitive
predicate
∈ on top of traditional
predicate calculus *without* equality. In
that case the Axiom of Extensionality becomes
(∀*w*(*w* ∈ *x* ↔
*w* ∈
*y*) → (*x* ∈ *z* → *y* ∈ *z*)),
and
equality *x* = *y* is *defined* as ∀*w*(*w* ∈ *x* ↔ *w* ∈ *y*).
All
of the usual axioms of equality then become theorems of set theory.
See, for example, Axiom 1 of [TakeutiZaring] p. 8.
To use the above "equality-free" version of Extensionality
with
Metamath's logical axioms, we would rewrite ax-8 1402
through ax-16 1606 with
equality expanded according to the above definition. Some of those
axioms could be proved from set theory and would be redundant. Not all
of them are redundant, since our axioms of predicate calculus make
essential use of equality for the proper substitution that is a
primitive notion in traditional predicate calculus. A study of such an
axiomatization would be an interesting project for someone exploring the
foundations of logic.
*General remarks*: Our set theory axioms are presented using
defined
connectives (↔, ∃, etc.) for convenience. However, it is
implicitly understood that the actual axioms use only the primitive
connectives →, ¬, ∀, =, and ∈. It is
straightforward to establish the equivalence between the actual axioms
and the ones we display, and we will not do so.
It is important to understand that strictly speaking, all of our set
theory axioms are really schemes that represent an infinite number of
actual axioms. This is inherent in the design of Metamath
("metavariable math"), which manipulates only metavariables.
For
example, the metavariable *x* in ax-ext 1877 can represent any actual
variable *v1*, *v2*, *v3*,... . Distinct variable
restrictions ($d)
prevent us from substituting say *v1* for both *x* and *z*. This
is in contrast to typical textbook presentations that present actual
axioms. In practice, though, the theorems and proofs are essentially
the same. The $d restrictions make each of the the infinite axioms
generated by the ax-ext 1877 scheme exactly logically equivalent to each
other and in particular to the actual axiom of the textbook
version. |