Re: Simplifying M theories.

On Jun 22, 1:04 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Jun 19, 5:40 pm, zuhair <zaljo...@xxxxxxxxx> wrote:
What I mean by axioms of ZF-Regulariy-choice is when we apply them to
"sets" in this theory.
Perhaps you mean the axioms relativized to the predicate 'set'. But
(1) that can't be taken as having very much to do with the ZF axioms,
since your predicate 'set' is defined in terms of a primitive that is
not in the language of ZF.
All of this is disappointing, since I had thought that you had turned
a corner away from those kinds of mistakes and over-claiming a while
ago.

OK, I think I see what's going on here. Indeed, it appears that zuhair
is making the same argument as Srinivasan was back in yet another
sci.logic thread. Let me see whether I can determine what exactly is
going on in both of these posts. Hopefully, MoeBlee won't consider
this to be a "waste of time," as he does with most of my posts. But
here I do not claim to have a proof that any theory is inconsistent.

Both Srinivasan and zuhair are trying to discuss two theories at the
same time, each theory having its own models and universes, and
this is what's confusing both sides of the debate. But let me give an
easier example to begin with.

Let PA be the theory of Peano Arithmetic.
Let RA be a theory whose axioms characterize the real numbers.

Now here's a simple question. For every model S of RA, does there
exists a model O of PA such that O is _embedded_ in S (where I
use the word "embedded" the same way that Fred Jeffries used the
word in the old Srinivasan thread)?

That is to say, let S be a model of SA. Let R be the corresponding
universe, so that S maps predicates of RA to subsets of R.

So we ask, does there exist a subset of R that is the universe of a
model of PA? Of course there is, since _any_ countable set can be
the universe of a model of PA, as long as the primitive symbols of
PA are mapped to the right relations. But what we mean is, is
there a model O of PA such that the primitives "0", "successor",
and especially "+" and "*", are "preserved" (i.e., O maps each
symbol to the intersection of the universe of O with each of the
images of the respective symbols via the model S)?

Of course there is. Just let N be the subset of R that corresponds
to the natural numbers (i.e., S maps the predicate "natural number"
to the set N). Then N is the universe of a model O of PA that does
preserve the "+" and "*" of the model S of RA. So even though any
countable subset of R can be the universe of a model of PA, only
the subset N preserves "+" and "*". It is therefore the only model
of PA that's _embedded_ in the model S of RA.

So it should be obvious that for every model of RA (the axioms
characterizing the real numbers), there exists a unique model of
PA that is embedded in the model of RA. Colloquially, we can
say that the real numbers "contain" PA.

If MoeBlee's going to argue this point, then I admit that reading
the rest of this post will be a waste of time. Also, notice that I
don't claim vice versa -- that every model of PA can be embedded
in a unique model of RA (which I believe is wrong). I only claim that
every model of RA has a unique model of PA embedded in it.

Now zuhair and Srinivasan aren't discussing PA or RA -- they are
discussing set theories. So let me give a simpler example of the
type of set theory they are discussing:

Let T be a set theory with two primitive symbols -- the usual "e"
and a one-place predicate "M", which can mean "is a set" (since
MoeBlee himself has used the symbol "M" to mean "is a set" in
the context of proper class theories in older threads).

Now here are the axioms. We begin with an axiom resembling the
standard Axiom of Extensionality in ZFC:

ZFC Extensionality:
Ax (Ay (x=y <-> Az (zex <-> zey)))

Our new version of Extensionality will only work for _sets_ (i.e.,
objects satisfying the predicate M). So we write:

T Extensionality:
Ax Mx -> (Ay My -> (x=y <-> Az Mz -> (zex <-> zey)))

In English, this says that two _sets_ are equal iff every _set_ that
is an element of one is an element of the other. (It says nothing
about when two objects that aren't sets are equal.)

Now let's move on to the Empty Set. Once again, we take the
axiom in ZFC:

ZFC Empty Set:
Ex (Ay ~(yex))

and modify it so that it refers to sets, as:

T Empty Set:
Ex Mx & (Ay My -> ~(yex))

In English, this says that there is an object that contains no sets
as elements, and this object is a set.

Similarly, we try Pairing and Union:

ZFC Pairing:
Aa (Ab (Ex (Ay (yex <-> y=a v y=b))))

T Pairing:
Aa Ma -> (Ab Mb -> (Ex Mx & (Ay My -> (yex <-> y=a v y=b))))

ZFC Union:
Aa (Ex (Ay (yex <-> Ez (yez & zea))))

T Union:
Aa Ma -> (Ex Mx & (Ay My -> (yex <-> Ez Mz & (yez & zea))))

And so on with all of ZFC's axioms and schemata. Every time we
see "Ay phi" we change it to "Ay My -> phi", and every time we
see "Ex phi" we change it to "Ex Mx & phi". We have relativized
all of the quantifiers to the _sets_, the objects satisfying M.

The resulting axioms and schemata are the axioms of T.

What can prove in this theory T? We observe that all of the
objects we normally discuss in the context of ZFC, such as 0,
{0}, omega, and so on, are sets (i.e., M0, Momega, etc.). In
fact, this theory neither proves nor disproves that there exist
_any_ objects that _aren't_ sets. Both "Ax Mx" and its negation
are independent of T. And notice that T+"Ax Mx" is obviously
equivalent to the theory ZFC+"Ax Mx". So it appears that
writing this theory with the symbol "M" is pointless and that
we might as well have just used ZFC without the symbol "M".

But here's the point I'm trying to make. The theory T has
many models -- some in which "Ax Mx" is true, and others in
which its negation "Ex ~Mx" is true. But for every model of T,
there exists a unique model of ZFC embedded in it -- and the
universe of the model of ZFC is the set to which the model
of T maps the symbol "M" -- a subset of the universe of the
model of T. If "Ax Mx" is true in the model, then the subset is
the improper subset, but if "Ex ~Mx" is true, then this is an
improper subset of the universe. And these are the more
interesting models of T -- and they are the models that both
zuhair and Srinivasan are trying to discuss.

One way to find a model of T in which "Ex ~Mx" is to take a
model of NBG, and let T map "M" to the image of "is a set"
via the model of NBG (i.e., the objects that satisfy M are
the sets, and the objects not satisfying M are the proper
classes of NBG). This is the most obvious way to find a
model of T in which "Ex ~Mx" is true.

Here's a less obvious model: Take a model of the theory
ZFC+inaccessible, and let T map "M" to the image of "has
cardinality strictly less than all inaccessibles." So the
objects that are sufficiently small are the _sets_, and the
objects that are too large (i.e., at least as large as the
smallest inaccessible) don't satisfy M. Then this is a model
of T in which "Ex ~Mx" -- and it is exactly the model that
zuhair is trying to describe! So when zuhair writes:

What I mean by axioms of ZF-Regulariy-choice is when we apply them to
"sets" in this theory.

he means that a unique model of ZF-Regularity is embedded
in every model of his theory -- and the universe of the embedded
model's is exactly the image of "is a set" via his theory. And so
MoeBlee's response:

(1) that can't be taken as having very much to do with the ZF axioms,
since your predicate 'set' is defined in terms of a primitive that is
not in the language of ZF.

is irrelevant. I've shown that the theory T contains a predicate "M"
that isn't in the language ZFC either, yet that doesn't prevent
every model of T from having a unique embedded model of ZFC. The
predicate "M" only tells us exactly which subset of the universe of
T's model is the universe of ZFC's model -- indeed, it's exactly the
image of the predicate "M" via T.

Here's yet another model of T. We begin with a model of IST, and
then let T map "M" to the image of "standard" via IST. Then this
should yet another model of T. We should be able to verify this
by replacing "M" in all of my axioms of T with "standard," then
proving all of the resulting formulas as theorems in IST. I believe
that they should all check (i.e., standard sets containing the same
elements are equal, 0 is standard, if a and b are standard then so
is {a,b}, etc.). And this is obviously the model of T that -- you
guessed it -- Srinivasan is discussing.

Every model of T (including these models of IST) should contain a
model of ZFC embedded in it. In the other thread Jeffries proved
that vice versa doesn't hold -- that not every model of ZFC is
embeddable in a model of IST. But I still believe that the reverse
is true, that every model of _IST_ has a model of ZFC embedded
in it, and the universe of the model of ZFC is the image of the
predicate "standard" via the model of IST.

In each case, we're discussing two theories at once -- T and ZFC,
NBG and ZFC, M+ (zuhair's theory) and ZF-Foundation, and
finally IST and ZFC. In each case, each theory has its own models
and universe, but we show that in each case, every model of the
.

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