Re: Small Set theory:Revised.


zuhair schrieb:

hagman wrote:

Now I have a question, should I add an axiom stating that for every set
there is a P that defines it.

How would you proof ~xex without it?
If one wants to use Ax.3 to show that ~xex, one needs that x is
P_embedded (shudder).

Why, all what we need to apply 3. is x is P_defined.

Exactly. I meant to say that *without* an additional axiom that every
set is P_defined for some P, you cannot simply apply ax.3 to prove ~xex
holds for all x.
But *if* you add that axiom, your first theorem (universe) becomes
false because ~vev follows.

.

Relevant Pages

  • Re: Small Set theory:Revised.
    ... Axiom 1 says that for any set there exists a formula P such that x is ... assume xex and let P be such that x is P_defined. ... if "y is not infinite" is ... In zuhair set theory, the Axiom Of Choice is probably not needed. ...
    (sci.math)
  • Re: Small Set theory:Revised.
    ... Axiom 1 says that for any set there exists a formula P such that x is ... assume xex and let P be such that x is P_defined. ... P((y is a subset of Power y and every member m of y is a subset ... In zuhair set theory, the Axiom Of Choice is probably not needed. ...
    (sci.math)
  • Re: k set theory
    ... Also I should add the axiom Ex ~xex ... I believe Ex ~xex should be sufficient. ... Null set: ExAy ~yex. ...
    (sci.logic)
  • Re: Naive Comprehension.
    ... Axiom schema of comprehension. ... member of itself, ... Then just prove ~xex, for the x that is in action in your proof. ... You haven't proven the existence of singletons, ...
    (sci.logic)
  • Re: Small Set theory:Revised.
    ... assume xex and let P be such that x is P_defined. ... But xex also implies Pby definition of P_defined. ... to what you already nicely presented, if yex then x!ey. ... (which is an axiom in ZFC) ...
    (sci.math)