# Re: on formally describable universes and measures

Here is an old reply to Russell Standish and Stephen Paul King.

Russell Standish wrote:

>I have often said myself the plenitude is not a set, however when
>trying to write up some of this work for another audience, I tried
>following up the web documents on set theory, I came up with nothing,
>so in the end simply didn't raise the issue.

I agree it is foolish to see the plenitude as a *set* and I have
explain why before.

Now, you can perhaps *modelize* the plenitude by a set, but *big* set
are notoriously difficult to handle.

So you need a set theory. Now, and this is a subtil point hardly
understood, a model of a set theory is
called *universe¨* and is intuitively a collection of *all*
possible sets. If we want to say "Group Theory" instead of
Transformation Theory we should say "Universe Theory" instead
of Set Theory!!!

In most set theories, (like ZF, NBG, ...), the universe is not
itself a set.
There exists set theories with "universal sets" in which the universe
is a set. The best known is NF (Quine's New Foundation), build from
some works by Church.

Well NF is not even known by some specialist of set theory, and
to study NF you need great familiarity with mathematical logic.

But sets are not enough. Even in mathematics there are mathematical
object which are to big to put in a universe (model) of ZF. For
exemples some categories.

Do you know category theory? It is mathematical
structure intermediate between group and lattices. They are bridge
between logic, topology, Algebra, etc.

Stephen Paul King wrote:

>    Oppps, I forgot to mention the notion of expressiveness...
>I am trying
>to keep my posts concise... Please
>read this paper by Peter Wegner which explains the notions of
>expressiveness and introduces Non-Well Founded
>sets, my thinking draws strongly from it:
>
> http://www.cs.brown.edu/~pw/papers/math1.ps

Formidable idea, let's do math. Non-Well Founded sets are certainly
interesting ... but it can lead us in the forest of mathematical
mermaids and keep us with some beautiful songs away from our search
toward a theory of everything.

It seems to me that many-worlder should so some modal logic, if only
to taste the second part of my thesis :-)
But even without my thesis I think that modal logic is a formidable
tool for rigorous philosophy. See my last post to George Levy.

Bruno

PS I have also problem to
I will try with the recent suggestion by Russell.