Re: Small Set theory:Revised.


zuhair schrieb:

zuhair wrote:
hagman wrote:
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.

This doesn't stop it from having a universe, since Ax.3 will forbid v
from being a member of any set, i.e v here will be the only proper
class in this set theory, while at the same time we have all other sets
as members of v. Therefore still the universe theorum holds.

Zuhair

if I put Ax.2 and Ax.3 in one axiom then this will vanish. I mean we
can have
a universe.

There's no difference between

Ax.1
Ax.2
Ax.3
Ax.4

and

Ax.1
(Ax.2 & Ax.3)
Ax. 4

.

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