FOM: re: Axiom of Extensionality

[Insall]

Dean Buckner wrote, on 17 May 2002:

`` What is this axiom?  Does it say that a set A is the same as set B iff
they have the same members?''

and

``...the set is one and the same with its members, in which case the axiom
reduces to {Alice, Bob, Carol} = {Alice, Bob, Carol}, and hardly seems
necessary.''


One thing the axiom of extensionality does for us is to distinguish between
ordered sets and unordered sets.  Thus, in the case you describe, namely
that {Alice, Bob, Carol} = {Alice, Bob, Carol}, the axiom is (perhaps)
unnecessary, as you say.  However, one other related application is

 {Alice, Bob, Carol} = {Alice, Carol, Bob}.

If I denote by parentheses [rather than curly brackets], a listing of the
members of an ordered set, then we have

 (Alice, Bob, Carol) = (Alice, Bob, Carol),

but the ordered sets (Alice, Bob, Carol) and (Alice, Carol, Bob) are
different.  This is because the axiom of extensionality only applies
directly to unordered sets.  (To ordered sets, it applies indirectly,
through various definitions of what an ordered set is, in terms of an
unordered set.)

The unordered version can be used to model situations like the equivalence
of the semantics of the English sentences

 ``Alice, Carol and Bob ran a race.''

and

 ``Alice, Bob and Carol ran a race.'',

while ordered sets can be reasonably used to model the inequivalent
semantics of the English sentences

 ``Alice won first place, Bob came in second, and Carol was third.''

and

 ``Alice won first place, Carol came in second, and Bob was third.''.


 Matt Insall