Re: ZFC in another shape.

On Feb 5, 5:06 am, "Rupert" <rupertmccal...@xxxxxxxxx> wrote:
On Feb 5, 3:00 pm, "zuhair" <zaljo...@xxxxxxxxx> wrote:





On Feb 4, 10:41 pm, "Rupert" <rupertmccal...@xxxxxxxxx> wrote:

On Feb 5, 1:02 pm, "Jesse F. Hughes" <j...@xxxxxxxxxxxxx> wrote:

"Jesse F. Hughes" <j...@xxxxxxxxxxxxx> writes:

"zuhair" <zaljo...@xxxxxxxxx> writes:

Hi All,

Is the following theory equivalent to ZFC?

Theory X.

Primitives: e,=

Definition Schema:

x is an ordinal <-> AmAn(((mex&nem)->nex)&Az(zen->zem)).

You sure that's the right definition? As consequence of this
definition, you get:

x is an ordinal -> AmAnAz(z e n -> z e m)

How is that? I think you are mistaken. See my answer to your previous
post.

No, he's correct. This is because from

P<->AmAn(Q&R)

you can infer

P->AmAnR

You really need a lot of practice in logic. For now it would probably
be best to accept that when more experienced people claim mistakes in
what you've done it's very unlikely that they're wrong. Just put your
mind to trying to understand what they're saying.

I only wanted to say that x is an ordinal iff
x is transitive and every member in x is transtive.

Okay, fine. Well, you didn't write it up correctly. That should be

x is an ordinal <-> AyAz((y e x & z e y) -> z e x) & Ay(y e x -> AzAw((z e y & w e z) -> w e y))

forgto to close it. But Beautiful. But I think it is long, couldn't be
writtin in a shorter manner.

Example.

x is an ordinal<-> AyAz( (yex & zey) -> (zex & Aw(wez->wey) ).

Is that equivalent to it.





which is a very well known definition of ordinals.

It's not obvious to me that this is equivalent to the usual
definition. The usual definition is that an ordinal is transitive and
connected, i.e. for all x and y in the ordinal we have either x e y or
x = y or y e x.

Hmmm... let me see how can I write this in symboles:

d is an ordinal <-> AxAy ( (xed & yed) -> (xey or x=y or yex ) ).

Am I correct?

I Know this, but the definition I mention and you properelly wrote it,
is proved to be equivalent to the standard definition Provided that
foundation
is an axiom.

Anyhow perhaps I wrote it in a mistaken way.

Yes, you did.- Hide quoted text -

- Show quoted text -

So I will present this theory again after the corrections.

Theory X.

Primitives: e,=

Definition Schema 1 )

x is an ordinal <-> AyAz((yex & zey)->zex)&Ay(yex->AzAw((zey & wez) ->
wey))).

Axioms:-

1) Extensionality
2) Foundation
3) Ordinal Comprehension:

AdExAy(yex<->((yed \/ y=d) & d is an ordinal )).

4) Replacement
5) Infinity
6) Choice
7) Power.

All axioms except 3, are exactly as in ZFC.

Now Is this theory equivalent to the standard ZFC?

Though I do think that "Power" can be a theorum in a theory made from
the first six axioms, but nevermind that, since I don't have a proof
now for it, I axiomatized Power as above.

Again the basic question here, is this theory equivalent to the
standard ZFC?

Zuhair



.

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