[FOM] FOM: Set Theory for Crocodiles

[Dean Buckner]

I wrote

>> ...  set theory assumes a set is not identical with its elements.
>> {Alice} is not the same thing as Alice, for instance.  {} has no members
to
>> be identical with.  And from the mere existence of Alice and Bob, we
cannot
>> infer the existence of {Alice, Bob}.

Neil Tennant wrote

> I thought set theory contained the pair-set axiom:

Indeed, but it's an axiom. It is required because a "set" in set theory, the
object denoted by the curly brackets, is not assumed to be identical with
its elements.  If what "{Alice, Bob}" denotes were the same as what "Alice
and Bob" denotes in ordinary English, we wouldn't need the axiom.

Neil wrote:

> Cantor's Theorem really belongs to the crocodile-brain part of set theory.

They must be clever crocodiles.  The general form of the argument is to show
for suitable c and R that

        (x) x R c iff  ~ x R f(x)

and to prove (by substituting f(m) for c and deriving contradiction)

    ~ (E m) f(m) = c

As I argued in a previous posting, we have to be sure "c" refers to
something.

Suppose we argued that a sentence like "Shergar is a crocodile" consists of
two names - "Shergar" and "is a crocodile", plus a relation that is asserted
when you put the two names together, call this relation "satisfies".  Then
put "satisfies" for R and "does not satisfy f(x)" for c.  Then

    x satisfies "does not satisfy f(x)" iff ~ x satisfies f(x)

That's not a very good argument.  For one thing, if we are saying that
every sentence of the form "x is F" is really of the form "x satisfies "is
F"" then we get an infinite regress, since this gives "x is an object that
satisfies "is F"" which itself is of the form "x is F".  (Bradley's
regress).

I don't see how the axiom of separation would get us out of this.  To avoid
regress, you need to show that

    {x in X: x not in f(x)}

signifies a bunch of objects, like the expression "Alice and Dave and Bob".
But it doesn't look like such a referring expression.  It looks just like a
predicate, dressed up in curly braces.

To connnect this with the first point about {Alice, Bob}, I wonder if we
could get a theory out the following assumptions.  Subjects terms signify
individual things.  These can be singular as in "Alice", plural as in "Alice
and Bob".  Of definite number but non-identifying as in "those two
crocodiles", of indeterminate number as "some crocodiles".  Predicate terms
by contrast signify nothing, but we can add them to subject terms to give
sentences.  Something along those lines.



Regards,

Crocodile Dean
The Thames
London
ENGLAND

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