Re: Let's you & him fight


george wrote:
Rupert wrote:
Never mind all this talk about sets and classes.

Jesus. Why must I endure a hijacking in the FIRST reply?!
I trust you will make this relevant later.


It is relevant and in fact I believe it is a good explanation of the
point Aatu was trying to make. Stop your pitiful whingeing.

Suppose I adjoin to
the first-order language of set theory a satisfaction predicate, the
idea being that Sat(n,x_1,x_2,...,x_k) holds when n is the Goedel
number

Jeezus. What does "adjoin" mean?

Read a dictionary.

It is kind of KEY to set theory that it ONLY has ONE predicate.
EVERYthing else NEEDS to be DEFINABLE. Is the predicate you
are adjoing definable? Is its definition finite?

I'm defining a new language. It is not the first-order language of set
theory, it is an extension of it. No, the new predicate is not
definable (as you are surely aware). Otherwise I would have said "I'm
defining a predicate, here is its definition." I'm extending the
language and introducing a new, undefined, predicate. I really don't
think my explanation was all that obscure.

And what is all this crap about Godel numbers? Is the
godel numbering definable?


My explanation of what the predicate Sat means was an informal
explanation designed to give you an idea of the intended interpretation
for the new language. It is not a rigorous definition since the basic
semantic notions cannot be defined, though they can be characterized by
axioms. To define Goedel numbers you would have to define formulas, and
of course if you are working in the first-order language of set theory,
you can never define the formulas themselves, only some surrogate for
them such as Goedel codes. I could define formulas by means of some
system of Goedel codes and then define the Goedel number of a formula,
but in that case all I would really be doing is defining a way of
translating between two systems of Goedel coding. When we're working in
the first-order language of set theory, Goedel codes are all we have.
You know how to work with Goedel codes as well as I do.

My explanation was a perfectly good explanation and I really think it
was such that a competent person could have understood it. But if
you're having trouble understanding it, don't get irritated and
confrontational, just politely ask for clarification.

of a formula in the first-order language of set theory with k
free variables

NO. I mean it. FORGET free variables.
You JUST PLAIN DON'T *need* them.


Sigh. If you want to understand what someone is trying to say, you
really have to listen to them. I won't be able to axiomatically
characterize the properties of truth unless I also have a notion of
satisfaction to talk about. For this I need to work with formulas with
free variables.

and it is true when these variables are interpreted to
be x_1, x_2, ... x_k.

You could just as well have allowed the formula to be
Ex1x2...xK[whatever], and said that Sat(n,x1,...,xK)
holds when the corresponding (fully grounded/instantiated)
existential is true.


Assuming I had a notion of truth to work with. Anyway, as a matter of
fact that's wrong.

Clearly, for all k, Sat_k(n,x1,...,x_k)=df=Sat(n,x1,...,x_k) is
definable.

Wrong. You obviously have no idea what you're talking about. So get rid
of the idea that you have something to teach me, and politely ask me
for help, and I'll see what I can do.

But is Sat finitarily definable? The "adjoining" and/or conjoining
of these infinitely many individual definitions is certainly not
a finitary definition.


It can't be defined. It's an undefined predicate. We introduce a new
predicate symbol for it.

I adjoin all the obvious axioms concerning the
satisfaction predicate

Is this finite?

Yes.

Is it recursive? Is it even quasi-finite via some
finite number of finitary schemata like replacement (as is normal
when axiomatizing ZFC)? And why are you doing this in ZFC
anyway? Why can't we do this in NBG?

Because the point of the exercise is to give an interpretation of NBG
in a language that we can all agree is meaningful without getting
involved in metaphysical disputes about whether classes exist.

The thread IS ABOUT
the set/class distinction.


I'm trying to show you that we can give NBG an interpretation in a
language which only uses semantic notions for the first-order language
of set theory, and doesn't use the concept of a class, so that you
don't need to resolve the issue of whether classes exist in order to
establish that NBG has a meaningful interpretation on which its
theorems come out true. I'm pretty sure that that's the point Aatu
wanted to make. I'm trying to explain it as I understand it.

and I assume that the axiom of separation and
replacement hold

Separation does not need to remain an axiom -- neither does
pairing, for that matter -- if you have Replacement.

True.

They are
theorems.

for formulas involving Sat as well. The resulting
theory, call it ZFC*, is not conservative over ZFC, it has much
stronger consistency strength. But surely if we accept ZFC we should
accept ZFC*.

I don't feel obligated to accept anything whose definition includes
"when these variables are interpreted to", especially not if it's
infinitary.

It's a perfectly well-defined recursively enumerable theory. I'm sorry
you're having trouble understanding its definition. As for the
philosophical issue of whether it should be accepted, maybe you can
come up with some decent arguments against my thesis that if we accept
ZFC we should accept ZFC*. But, given the level of understanding you've
demonstrated so far, I'm not filled with confidence.

Interpretations and models generally ARE NOT necessary to this whole
enterprise; that is the whole import of the Completeness theorem.


I didn't use any model theory. I gave a syntactic definition of the
theory and made some remarks which were meant to be suggestive about
its interpretation (that's not in a model-theoretic sense, since model
theory only deals with interpretations whose domain of discourse is a
set).

And NBG can be interpreted in ZFC*. That is, there's a
birecursive map from the theorems of NBG to a recursive subset of the
theorems of ZFC*.

How can THAT be possible when the theorems of NBG *by itself*
are NOT (totally) recursive? They are instead/merely recursively
enumerable.

Good point. This was the wrong definition. I amended it in a later
post.

Aren't the theorems of ZFC* also recursively enumerable?


Yes.

If we accept that talk about semantics of the
first-order language of set theory is meaningful,

Well, I personally don't think we should.
What's formalizable is what we're going to accept as meaningful.
We may meaningfully communicate about that semantics BEFORE
we formalize it, but nobody HAS to accept anything until AFTER it's
formalized.

I did formalize it.


then we can give a
meaning to the sentences of the second-order language of set theory on
which the theorems of NBG come out true.

Since NBG itself IS a set theory, this is an elephant laboring to
bring
forth a mouse. I can't believe anyone is surprised that given a
consistent
first-order theory, you can design an interpretation of the 2nd-order
theory
OVER THE SAME signature,

No.

WITH THE SAME 1st-order axioms,

No.

with
appropriate lifting of schemata, that preserves it.


I would be surprised if that claim is true, but it wasn't what I was
claiming.

The point of interest is that ZFC* is an intuitively acceptable theory
and we can interpret NBG in it.


> On the other hand, the same
probably couldn't be said of Kelly-Morse set theory.

I would REALLY appreciate it if people would STOP talking
about ZFC AND Kelly-Morse and just KNOW that THIS thread
is about NBG.

Well, I would really appreciate it if you would realize that you don't
own this newsgroup and that people will talk about what they damn well
want to and you can stop making complaints about it, thank you very
much.

And I certainly don't see why you needed to invoke
2nd-order anything if you were "adjoining" a first-order Sat predicate!

The only sense in which I "invoked 2nd-order anything" is that I talked
about NBG. You were the one who kept whingeing that the thread is about
NBG so I don't know what you're complaining about. My result is that
NBG is interpretable in ZFC*, that's the result I was explaining to
you. In my view, the philosophical interest of this claim is that we
can accept that the language of ZFC* is meaningful without relying on
any controversial thesis about the existence of classes. As I say, I
believe this was Aatu's point.

.

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