Dear John, (01)
> On 12/17/2010 10:03 AM, Matthew West wrote:
> > ISO 15926 does say that a class can be a member of itself for reasons
> > Chris P has alluded to, but it is not committed to a particular form of
> > non-well foundedness.
> As I have said before, I have no idea what a *class* is supposed to be.
> If a class is supposed to be a set, then it should be called a set. (02)
MW: The name class is part of standards history. (03)
> Therefore, I assume a class might be set-like, but it's not a set. (04)
MW: Well that begs the question: What is it to be a set? (05)
The normative part of the definition of class in ISO 15926 is as follows: (06)
A <class> is a <thing> that is an understanding of the nature of things and
that divides things into those which are members of the class and those
which are not according to one or more criteria. (07)
The identity of a <class> is ultimately defined by its members. No two
classes have the same membership. However, a distinction must be made
between a <class> having members, and those members being known, so within
an information system the members recorded may change over time, even though
the true membership does not change. (08)
A class may be a member of another class or itself. (09)
There is a null <class> that has no members.
END QUOTE (010)
If I had my time again I would argue more strongly for not including the
first sentence, because there is nothing to stop you using classes for
arbitrary collections. But I've seen worse definitions produced by
MW: On the whole I would be inclined to call something whose identity is
defined by its membership a set, but this clearly does not refer to a
standard set, though they will obey the usual set operations.
> If a class is not a set, then there is no reason why any particular
> axioms for sets should apply to it. The following statement (or its
> translation to any version of logic) does not, by itself, create any
> problems or contradictions:
> "A class may be a member of itself."
> The contradiction that arises in common versions of set theory is
> caused by the following axiom:
> For any monadic predicate P, there exists a set of all x
> such that P(x) is true. (012)
MW: Yes. I'm happy to drop that in favour of a set exists if its members
exist and can be enumerated (explicitly or implicitly for infinite sets). (013)
MW: Then if a predicate does not correspond to a set, then it is just
nonsense rather than a paradox.
> The contradiction arises with the following choice of P:
> P = (lambda x)(x is not a member of x).
> If you don't assume that some set (or class) must exist
> that is determined by this P, then there no contradiction.
> And by the way, Common Logic manages to avoid contradictions
> by a method similar to what I just described: the examples
> that create paradoxes in some versions of logic just produce
> CL statements in which the paradoxical entity doesn't exist. (014)
MW: And what do you call that version of set theory? (015)
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