Re: Post Axiom Syndrome

Chris Menzel wrote:
> On 10 Jul 2005 20:04:26 -0700, Ross A. Finlayson <raf@xxxxxxxxxxxxxxx> said:
> >> ...So, assuming you desire to be rational, you really ought to stop
> >> claiming that ZF is inconsistent until you have a proof *in ZF* that
> >> the universal set exists.
> > ...
> > A model is to a theory as a class is to a set. What is the set of all
> > sets? What is the class of all classes or group of all groups or
> > collection of all collections ad infinitum? ZF is inconsistent.
>
> Ok then, well, rationality isn't everyone's cup of tea, I guess.

Actually, what we're talking about is more along the lines of how
quantification over sets implies a set of all sets, in a set theory.

I have a special name for a set's complement in the set of all sets:
the context.

Consider the world with two kinds of things: apples, and oranges. If
the collection of all of those things does not exist, then neither does
one or the other because there would not be anything else.

Cantor, Burali-Forti, Russell, the paradoxes, they are similar things,
and basically regularity was added to naive set theory to skirt
discussion of such notions as the Ding-an-Sich, the set of all sets
that don't contain themselves, the order type of all ordinals, or the
set of all sets as its own powerset.

Those things simply "don't exist" in ZF, they're "undefined." In a
variety of other theories for which many of the theorems of ZF are
theorems, they do. We're just talking about them, ZF is quite mute on
the matter.

Basically, you either quantify over every object in the theory, or
don't. ZF cannot. Now, I understand that some people don't think or
understand why general quantification implies quantification, or
basically a choice function with a natural index, over each object of
the theory. There are others, obviously, whose opinion it is that
universal quantification is over each element of the universal set, in
a set theory.

In some ways these aspects are about the plain mechanics of theory,
here a set theory, they are things that have meaning for any rational
and comprehensive, ie towards general comprehension, treatment,
formulation, and application of those theories

ZF is obsolete. It's also inconsistent, or it's rational for you to
expect a universal set in a theory of sets, and it's incomplete. Don't
you think a theory to explain everything rationally need be able to
prove every true thing, ie be complete? Infinite sets are equivalent,
as the universe is infinite.

Ross

.

Relevant Pages

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