removing one set from another
Rodham E. Tulloss
ret at PLUTO.NJCC.COM
Tue Jun 17 12:30:25 CDT 1997
As a mathematician (in fact as a metamathematician who lived set theory in
grad school), may I suggest that Maureen Kearney is confusing the
membership of a set with the common properties of those members. In fact,
sets are defined by statements (all x, such that x is blue and has green
stripes) including, but not restricted to, the statement that lists
membership.
If I have a set of the "blue things with green stripes" and someone paints
a few of them red and transfers them to the set of "red things," my
original set (with reduced membership) is still the set of "blue things
with green stripes."
However, this example is still not quite right for
the point under discussion. For, in fact, what we are talking about is
something a little beyond classical set theory. The "blue things with
green stripes" can reproduce. Take an extreme case: Blue things with
green stripes are nearly extinct. There is a
set with only one blue thing with green stripes.
Luckily, it reproduces asexually. By its reproductive process
it produces offspring. The first time it reproduces,
something intervenes and it
produces red things. These will be classified in the set of red things.
The original set is still the same size and contains the same (single)
member. It's defining statement is unchanged even if one includes
cardinality of the set as a character. The red
things go their own way. Now, second case, the one
and only blue thing with green stripes reproduces again. This time only
blue things with green stripes are produced. The original "set" is now
increased in membership, but it is still the set of blue things with green
stripes.
If anyone is concerned about sets changing size, think of a set's
membership being enumerated. The list of things enumerated increases
in size. In fact when a set is defined
as the output of an automaton or is recursively
enumerable (sorry for the technical term), we know the properties of its
members, at least we can start listing them, but we cannot be sure that
we have them all because we don't know when or if the process of enumeration
will stop (sort of like whether or not a taxon will become extinct after
N million individuals have existed or after N million plus 1 individuals have
existed or until well after the death of the Sun).
Rod
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