Re: A set theory equivalent to ZFC.

On Feb 11, 3:47 am, "Rupert" <rupertmccal...@xxxxxxxxx> wrote:
On Feb 11, 11:07 am, "zuhair" <zaljo...@xxxxxxxxx> wrote:





On Feb 10, 6:13 pm, "Rupert" <rupertmccal...@xxxxxxxxx> wrote:

On Feb 11, 2:40 am, "zuhair" <zaljo...@xxxxxxxxx> wrote:

On Feb 10, 4:02 am, "Rupert" <rupertmccal...@xxxxxxxxx> wrote:

On Feb 10, 7:00 pm, "zuhair" <zaljo...@xxxxxxxxx> wrote:

Hi All,

The following is a set theory that I claim to be equivalent to ZFC.

---------------------------------------------------------------------------­­­­­­-----
-Small Set Theory-

This theory is an extension of first order logic with identity axioms
and special identity semantics too.

Primitives: e,=,-

"-" refers to juxtapositioning.

so x-y means x juxtapositioned to y.

Okay, I'll say a couple of things.

First of all, you appear to be giving a theory in a different language
to the language of ZFC. So before we can ask whether your theory is
equivalent to ZFC, we'll have to specify an interpretation of the
language of your theory in the language of ZFC. I haven't read all of
your post, so I don't know whether you do this later on, but certainly
you'll have to do it for your question to make sense.

It is the same first order language of ZFC.

No, your primitives are e, =, and -. The primitives in the first-order
language of ZFC are e and =. Until you come up with a definition for -
you're working in a different language.

Well in that case, I will demand a definition for "e" used in ZFC.
"-" is a Primitive, a primitive cannot be formally defined, can you
define "=", I didn't see any attempt to define it in ZFC. so why I
should define "-".

You are welcome to have whatever primitives you want. But it doesn't
mean anything to say two different theories are equivalent unless each
language can be interpreted in the other. If you are claiming your
theory is equivalent to ZFC, you must specify how to interpret the
language of your theory in the language of ZFC in order to give your
claim meaning. Either that or you might just claim your theory is
conservative over ZFC. (I.e. every sentence in your theory which is in
the language of ZFC is a theorem of ZFC, and conversely).

So what do you want to do? Do you want to claim the two theories are
equivalent? If so, you must specify an interpretation of the language
of your theory in the language of ZFC. Or do you just want to claim
your theory is conservative over ZFC?

Ok, Just let us focus now the case of the following theory being
EQUIVALENT to ZFC, and I will specify an interpretation of the
language of this theory in the language of ZFC.

Theory X.

primitives: e,=


Definition 1)

x is an ordinal <-> AyAz(zey&yex->(zex & Au(uez->uey)).

Axioms: 1) Extensionality 2)Regularity 3) Pairing 4) Union 5) Infinity
6) Axiom schema of Relation 7) Axoim schema of Size Comprehension 8)
Power



Were 1,2,3,4,5,8, are as in ZF


The axiom schema of relation is


AaAbExAy(yex<->( (y=<z,u> & zea&ueb) & P(y) ) ).


were <z,u>={{z},{z,u}} , i.e the standard definition of ordered pair.


P is a formula in first order language in one free variable y, that
doesn't use the symbole x.

what this axiom is saying is that for any two sets a and b, any subset
of the cartesian product axb Exists. And therefore from this axiom and
from pairing we can define relations , functions, injections,
surjections ,bijections,..etc.


and axiom schema of size comprehension is:


ExAy( yex<->(P(y) & Ez( z is an ordinal & |z|=|x| ) ) ).


were |z|=|x| according to the standard Cantorian definitions, i.e

|z|=|x| <-> (Ef(f:z->x & f is injective) & Eg(g:x->z & g is
injective).

and P is a formula in first order language in one free variable y,
that
doesn't use the symbole x.


.........................................................................

-Interpretation of the language of this theory in the language of
ZFC.-

Now: This theory is an extension of first order logic with identity,
and thus uses the same language of ZFC.The only two axioms that are
not in ZFC, is the schema of Relation and schema of size comprhension.

Now schema of relation is a theorum in ZFC.

what remains is size comprehension.

The idea of size comprehension is to construct an axiom in this
theory, that works like the axiom of limitation of size in NBG, and
thus it entails replacement, separation and choice in one go.

Since size comprehension requires for any x to be bijectable to an
existing ordinal, then this is equivalent to saying that x is not
bijectable
to what is known as proper classes in NBG. and accordingly I expect it
to play the same rule as limitation of size plays in NBG.

Therefore I claim that this theory is equivalent to ZFC.

Am I right?

Zuhair





By the way, this primitive -, is it a predicate symbol or a function
symbol?- Hide quoted text -

- Show quoted text -


.

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