Re: Proper classes


Jesse F. Hughes wrote:
> "Jonathan Hoyle" <jonhoyle@xxxxxxx> writes:
>
> > In non-Well Founded theories, where K can be a set, you have a
> > different problem. To be a set with cardinality, K must be in
> > one-to-one correspondence with some cardinal number. This cardinal
> > number must, in turn, be a member of the K. Since the K contains
> > *all* cardinals, including itself, it must itself be "the largest"
> > cardinal, since K+ = K U {K} = K. We are left with all sets are of
> > cardinality <= K.
>
> By definition, cardinal numbers are ordinals and hence well-founded.
> Even in set theories with an anti-foundation axiom.

Jesse, I disagree that "by definition" cardinal numbers are ordinals.

By definition, cardinal numbers are equivalence classes of sets,
the equivalence relation being that of set isomorphism. (If you
assume axiom of choice, the distinction becomes moot, but
more on that in a moment.)

In Cantor's paradox the natural "candidate" for largest cardinal
is not the cardinality of all cardinals, but rather the cardinality
of the universal set.

I can speak to how this "paradox", qua Kant, of both largest
and no largest cardinal numbers plays out in Quine's NF, or
equivalently JB Rosser's ML (NF + proper classes).

The cardinal number of the universal set is well-defined, and
it is indeed the largest cardinal number.

The power set of the universal set is again the universal set,
and again this well-defined.

What fails to happen is the construction of the "diagonal"
that shows strict inequality between cardinality of a set and
its power set. The axiom scheme of comprehension is
restricted in NF/ML to "stratified" formulas, and as is pretty
well known, when Cantor's diagonal argument is applied to
the universal set in this connection, it reduces simply to
Russell's paradox. So really what has to be noticed here
is that the property of a set not being an element of itself
is not expressable as a stratified formula.

[A stratified formula is one in which "ranks" may be given
to all the variables in a manner that atomic formulas like
"X in Y" occur only if the rank of X plus 1 is the rank of Y.]

The Burali-Forti paradox, of largest and no largest ordinals,
plays out in similar but arguably more interesting fashion.
Recall that in this instance the natural candidate for largest
ordinal is the (equivalence class of the) class of all ordinals
(ordinals defined as equivalence classes of well-ordered sets,
with equivalence relation being that of order isomorphism).

The set of all ordinals does turn out to be well-defined,
and it is itself well-ordered in NF/ML (any 2 well-ordered
sets are comparable by order-preserving monomorphism).
However any ordinal may be enlarged by "adding one"
(successor ordinal). So the (equivalence class of) the set
of all ordinals is not the largest ordinals, and the naive
attempt to "embed" an arbitrary well-ordered set as the
initial segment of "all ordinals" runs afoul of the restricted
axiom scheme of comprehension previously mentioned
(stratification).

There is a kind of tragic/comic ending. Using some
ideas related to these, Ernst Specker showed in 1953
that the axiom of choice is false in NF (equiv. ML). But
the silver lining is that the "axiom" of infinity is provable,
something that is assumed "ad hoc" in ZFC/NGB.


regards, chip

.

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