[FOM] Sets and Proper Classes
Robert Lindauer
robbie at biginteractive.com
Mon Feb 23 12:33:05 EST 2004
>> A set is an arbitrary collection of objects.
>> A property of sets
>> for which there is no set of all sets that satisfy the property is
>> known
>> as a proper class.
Can you explain what you mean by "arbitrary" and how it is to be
distinguished from "any old ... whatsoever" and why this wouldn't
include the simple predicate "everything" or "all" as in the extension
of the inverted A in (standard?) logic, eg: "A(x) x = x" or " A(P)
P<->P", etc. (Here we'll have to appeal to Wittgenstein's "fact-world"
in order to ensure that the second is always true, I'm not sure how I
feel about it - in any case, if you can't express it, it's not going to
be governed by the rules of syntactic logic anyway.)
That is, the force of "arbitrary" and "collection" must have their
meaning pre-theoretically if they are to have the force being put on
them here. Otherwise, they seem to be restricting what has already
been claimed to be arbitrary. If "arbitrary" has a technical meaning
for you - e.g. doesn't mean what we innocent onlookers think it does,
that technical meaning will not be anything like what it is in English.
Once you invoke "arbitrary" in English, anything goes. "His wives are
arbitrary" might mean that he's married a dog.
Perhaps the better way of saying "arbitrary" in this case is: "If we
call everything a set, then the set of everything has a cardinality,
according to our normal rules, which is higher than itself and lesser
than itself. We therefore call things that aren't everything "sets",
and things that are "proper classes". " And therefore take on the
immediately axiomatic attitude.
The alternate desire to retain some semblance of "intuitiveness" is
mistaken - who would you be kidding?
>> We know that for every set s, there is the set of all subsets of s.
"We" take this on faith or stipulate it, it is an axiom. I don't know
what the sense of the word "know" is when applied to axioms. This is
like "I know how to speak English" not "I know that my car is in the
Garage." How is it like the one, not the other? The contrary isn't
just not true in the language in question, it's simply nonsense in the
language in question. "I don't know how to speak English" is like "The
powerset axiom is false in ZFC". Is it even meaningful outside of
ZFC? In what sense is it the "same" axiom if the rules for how it is
applied are all different - same in some ways, different in others.
Again, the problem is mixing the pre-theoretic "know" with the
axiomatic "powerset" to derive the appearance of having
"known-in-English" that the powerset axiom is true when in fact what we
know-in-English is that it is an axiom in some axiomatic systems and
not others and has such-and-such expression and has these consequences
in this system, but not in some others, etc.
Best Wishes,
Robbie Lindauer
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