Re: The Dedekind Snap

On Dec 16, 2:58 pm, Herbert Newman <nomail@invalid> wrote:
On Tue, 16 Dec 2008 14:41:29 -0800 (PST) MoeBlee wrote:

Note that in this case "0" is a primitive.

No, it is defined.

???

I'm sorry to point out, but you're doing that irritating thing again
of responding line by line rather than reading my post with at least
enough context to get to the next line before you go "???" or
whatever.

We prove: E!xAy ~yex.

Then define: 0=x <-> Ay ~yex.

Ah. Then you will not have any urelements in your theory (i.e. effectively
excluded them), you know, right. :-)

I think "Z set theory" may be understood to have no urelements, as
opposed to "Z set theory with urelements".

But I agree with this: To ENSURE that there ARE urelements, it seems
one needs to take an additional primitive other than 'e', whether that
additional primitive is 'is a set' or '0'.

So why not just define

        set x :<-> x = x

in this case?

Because, though that is a theorem of Z set theory, it is not a theorem
of NBG, so I like the idea of having a defintion of 'is a set' that is
portable to certain class theories too.

c is a class <-> (Ex xec v c=0)

Well, no proper classes in Z (sic!) set theories! ;-)

Yes, so?

And why the 'sic'?

Because "Z" (->Zermelo) set theories only know about /sets/ (and possibly
urelements). (Hence it's not sensible to define /proper classes/ _in this
context_.)

What? Come on, you very well know that even though we prove that there
do not exist proper classes, it is not disallowed to define a
predicate symbol 'is a proper class' and then prove that nothing
satisfies that predicate. The definition is perfectly allowed. For
definition of a PREDICATE, it is not required that there be an object
that satisfies the definition. It is perfecty rigorous formal theory
to define a predicate (irrespective of whether that predicate can be
satisfied) and then prove or not prove that there it is a predicate
that holds of no object. You may not like it heuristically - that's a
matter of taste - but it is formally allowed.

Remember your original claim was:

        "One can define 'set' in such theories as Z set theories."

With other words:

  y is a set <-> (Ex xey v y=0)

No, that's NOT my definition.

My definition is:

y is a set <-> (y is a class & Ex yex)

So

y is a set <-> ((Ex xey v y=0) & Ex yex)

It's obvious at this point that you didn't really read my previous
post.

Now from your theorem above:

        E!xAy ~yex

and the definition

        0=x <-> Ay ~yex

We get the theorem:

        Ax(set x).

Actually, we get

Ax x is a set

from the pairing axiom or from the power set axiom, whatever you
prefer. (And extensionality to prove there are no urelements.)

Well great. That's why I propose the definition

        set x :<-> x = x

in this case. Then we still get the theorem

        Ax(set x).

And you just skipped again my reason for not adopting that definition.
I like having a definition that works both in Z and in NBG.

On the other hand, if you intend that your theory allows for urelements,
and you want to use the definition

        y is a set <-> (Ex xey v y=0) ,

No, that is NOT what I want. And if I want not to exclude that there
may be urelements, then 'is a set' or '0' will be primitive. Say, '0'
is primitive'.

Then again:

y is a set <-> ((Ex xey v y=0) & Ex yex)

So my definition is portable not just to NBG but also to Z allowing
urelements.

"0" "most probably" is a primitive of your system. (Just what I claimed.)

No, either '0' or 'is a set' would be primitive if I wanted not to
exclude that there may be urelements. But by "Z set theory" I mean
ordinary Z set theory that excludes urelements (such as an ordinary
textbook axiomatizaton (though there are exceptions, e.g., Suppes, who
does allow that there may be urelements)).

A very reasonable definition imho: Something is a set if it has some
elements,  o r  (otherwise) is the empty set (sic!).

And that doesn't work in NBG. So, I say something is a set iff it has
elements (or is the unique non-urelment not having an element) AND is
itself an element of something TOO.

Meanwhile something is a CLASS if it has elements (or is the unique
non-urelment not having an element). And something is a proper class
iff it is a class but not a set.

MoeBlee
.

Relevant Pages

  • Re: The Dedekind Snap
    ... In Z set theory without urelements (what we ordinarily ... NOT DENY that I use '0'. ... other primitives than 'e' and '=') in Z set theory WITH possible ...
    (sci.logic)
  • Re: urelements and the natural numbers
    ... the successor function with set operators and we're ... in ZF"everything is a set" (i.e. no urelements). ... naturals in the lambda calculus. ... the axioms of set theory, we simply do not need to add all those stuff ...
    (sci.math)
  • Re: Implementable Set Theory and Consistency of ZFC
    ... ZFC may have different variants: We can have a variant in which there ... are urelements and a variant in which it is undetermined whether there ... My implementable set theory is the one without ...
    (sci.math)
  • Re: urelements and the natural numbers
    ... the successor function with set operators and we're ... Since we presumably have no operations on urelements ... set theory and/or formal logic existed - I believe his work ... other essay is well worth reading, ...
    (sci.math)
  • Re: urelements and the natural numbers
    ... the successor function with set operators and we're ... urelements with set operators (other than to form sets ... the naturals, or else just talking about the naturals by ... about them as urelements embedded in a set theory ...
    (sci.math)