Re: Small set Theory:final version.


Jesse F. Hughes wrote:
"zuhair" <zaljohar@xxxxxxxxx> writes:

zuhair wrote:
I think ZFC can be translated to the following:

Premitives e

Definitions:

x is P_not embedded <-> Ay( yex<->P[y]).

Axioms:

1) Extensionality: As in ZFC.
2) Regularity: As in ZFC.
3) Comprehension: AP( (ExAy(yex<->P[y])) <-> ~P[x] )
were P is a predicate in one variable that doesn't use x,
nor it is biconditioned to a predicate in one variable that use x.

Once again, you have x free on the right hand side and bound on the
left. What a strange axiom!

ah, yea you are right,this is wrong then, in reality this is was not my
original idea.

correction:

3) Comprehension: AP( (ExAy(yex<->P[y])) <-> x is P_not embedded ).

Now it doesn't matter if x on the right of the biconditional is free,
since for every P there
is one and only one set that is defined after it.

However I know that even with this axiom , this theory though It
resembles ZFC. but it is not equivalent to it.

see that for example the set of all transitive sets which is not a
transitive set, is a set in this theory, while it is not in ZFC.

Zuhair

--
Jesse F. Hughes
"Radicals are interesting because they were considered 'radical' by
the people who played with them who wrote a lot of math work that
modern mathematics depends on." --Another JSH history lesson

.

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