Dear Matthew and Duane, (01)
MW> It is the most basic thing about sets that they are defined
> by their membership, which does not change. (02)
Yes. If you write something like the following, (03)
S1 = S union {x} (04)
mathematicians don't say that S has changed, but that S1 is
a different set. (05)
In programming languages, it is common to write (06)
x = x + 1; (07)
but when that operation is carefully analyzed and described,
it's described as "the value stored at the location designated
by x has been replaced or updated with a new value." (08)
There are also some programming languages, such as ML or Haskell,
called *functional* or *single-assignment* languages in which
no variable can be reassigned a different value -- i.e., no
statement of the form x=x+1 is permitted. (09)
MW> So if you have something that has members, but the membership
> can change, then what you know for certain is that it is not a set.
> Some people use the word type for such things. A type will have,
> at a point in time, a set which is its membership. (010)
The usual distinction: (011)
The identity conditions for a set are *extensional*: if S1 and S2
have the same members, they are identical; otherwise they are two
different sets. (012)
The identity conditions for a type are *intensional*; two types
T1 and T2 are identical if their definitions are equivalent;
their set of instances in a particular world w is called their
*denotation* in w. Those worlds may be, for example, the physical
world or a particular computer system at different times. (013)
John (014)
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