[Phil-logic]Cantor's Rule (was pairs)

[Robbie Lindauer]

On Jul 5, 2004, at 11:39 AM, KRamsay at aol.com wrote:

>    We will call by the same "part" or "partial aggregate" of an
>    aggregate M any other aggregate M_1 whose elements are also
>    elements of M.
>  
>    If M_2 is part of M_1 and M_1 is part of M, then M_2 is part
>    of M.
>  
> So in the previous sentence Cantor defines the word, whatever it is,
> to mean what we mean today by "subset".

Oy vey.

The context of this discussion was consideration of a proposed 
"logical" definition of, say,

3  == E(x,y,z): x!=y&y!=z&x!=z & An n=x|n=y|n=z

That is, the number of things in that domain.  This is fine as far as 
it goes.  This is what I said at the time.  It doesn't work with sets 
though for two main reasons which I tried to explain last year:

1)  If x, y or z are all sets, then more things are implied to exist 
than just those three sets by the truth of the expression.
2)  If x, y or z are all ONLY sets, then at least one of them is the 
empty set, and the choice to say that the empty set is 1 or no things 
is completely arbitrary.

My invocation of Cantor's expression was simply to show that if "M2 is 
a part of M1 and M1 is a part of M and M then M2 is a part of M" then 
surely if M2 is a part of M and M exists, then M2 exists.

This is to say, if (3) is true, then the objects over which it ranges 
are not sets.  It's a forgone conclusion.

The point of saying such things is this - that if this is the 
DEFINITION of 3, then there is no such thing as 3 sets OR at least one 
of them is a urelement and the decision to call this anything is 
absurd.

This was in response to the question of whether or not the Cardinality 
Axiom or its equivalent was obvious or true or provable or anything of 
the sort.

____________________________

The other issue is the univocality of "member of" or "is in".

I recognize that there is a difference between the expression "in" in 
English which tends to have a transitive meaning:

"If Robbie is in the submarine and the submarine is in the ocean then 
robbie is in the ocean."

There are similar expressions for sets along the lines of Cantors Rule:

"If M2 is a part of M1 and M1 is a part of M, then M2 is a part of M".

BUT, this means that there are at least two kinds of "being in" since 
the idea that they are univocal leads to contradictions (the ones I 
pointed out last year).

But then the question becomes whether or not there just are 
contradictions and we are making things up in order to avoid them OR 
whether the univocality of "being in" is the source of the 
contradiction.

Consider the following groups (not to say SETS):

{ {}, {{1}}, {{1}, {}} }

By Cantor's rule, 1 "is a part of" this set.

Note that it would be perverse to say that by Cantor's Rule

{ {}, {{1}} } is a part of the set

Since it is not a member of any of the members of the set.  It is, 
however, a subset of the set.  I would say that this is definitive 
against interpreting the expression as our modern "subset" expression.

_____________

Note also that this expression is important for cardinality, not for 
set relations.  The reason it's important for Cantor's consideration of 
cardinality is that there is a question - what is the cardinality of a 
given arbitrary set?

Since, for him, it is axiomatic that there is a cardinality for any 
arbitrary set, there has to be a definite way of determining what it 
is.

When we have complicated sets:

{1, 2, 3, 4, ..., aleph-0, aleph-2}

There is a question, "What is the cardinality of the set?"

Well, it depends on how we count and what we count.

If we count aleph-0 as just one thing and aleph-2 as just one thing, 
then the number of things in the set is aleph-0.  If we count aleph-0 
and aleph-2 as their constituent members (and if 1, 2, 3, 4 "are sets" 
and we count their constituent members) then there are aleph-2 things.  
If we think it perverse not to identify a thing with its unit set like 
"{} = {{}}" (a set of nothing is identical with a set of a set of 
nothing - it is nothing), then we might say that there is nothing in 
the set at all.  If we think that the urelement is my cat, then we 
might say there is one thing (since surely my cat has no subsets or 
members in the relevant sense).

|{{{{{{{robbie's cat}}}}}}}| = ?

The point is just to point out the places where decisions are to be 
made (and were made) and how alternate decisions might be made and how 
they (might) proceed - pointing out that there is no reason why they 
shouldn't proceed that way except to rescue some a priori features of 
the system which the system-designer is trying to preserve.

In particular, I think Cantor was trying to reproduce the structure of 
the natural numbers and the real numbers by modeling them with sets.  
He already had a good idea of what the reals and naturals were.  This 
activity is wholly superfluous given the existence of the natural and 
real numbers (in fact, given the existence of the natural numbers).

Boshuck likes to say that without producing an equivalently good 
system, there's not point in criticizing the entrenched one.  But this 
is a scientific mistake.  Good things come from a variety of sources in 
science.  In any case, it has been pointed out that the real number and 
natural number systems are infinitely more simple than their set 
theoretical counterparts and lead to only slightly fewer paradoxes and 
lots fewer decisions to be made in the process.  The existence of other 
formal systems suitable for modeling the reals with natural numbers 
leads us very simply back to where we started - the natural number 
system.

This is felicitous, since it is these numbers that we need for doing 
the rest of our sciences and ontology

Best,

Robbie Lindauer