ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax-ext Structured version   Unicode version

Axiom ax-ext 2008
Description: Axiom of Extensionality. It states that two sets are identical if they contain the same elements. Axiom 1 of [Crosilla] p. "Axioms of CZF and IZF" (with unnecessary quantifiers removed).

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 , and equality is defined as . 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 1382 through ax-16 1683 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.

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 in ax-ext 2008 can represent any actual variable v1, v2, v3,... . Distinct variable restrictions ($d) prevent us from substituting say v1 for both and . This is in contrast to typical textbook presentations that present actual axioms (except for axioms which involve wff metavariables). In practice, though, the theorems and proofs are essentially the same. The $d restrictions make each of the infinite axioms generated by the ax-ext 2008 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
ax-ext
Distinct variable group:   ,,

Detailed syntax breakdown of Axiom ax-ext
StepHypRef Expression
1 vz . . . . 5  setvar
2 vx . . . . 5  setvar
31, 2wel 1381 . . . 4
4 vy . . . . 5  setvar
51, 4wel 1381 . . . 4
63, 5wb 98 . . 3
76, 1wal 1231 . 2
82, 4weq 1379 . 2
97, 8wi 4 1
Colors of variables: wff set class
This axiom is referenced by:  axext3  2009  bm1.1  2011  dfcleq  2020  a9evsep  3855
  Copyright terms: Public domain W3C validator