Re: Z.Irregular Set Theory


Ross A. Finlayson wrote:
zuhair wrote:
- Irregular Set Theory-


Primitive e


1) Axiom of Extentiality: Ax,Ay: x=y <-> (Az:zex <-> zey)
2) Axiom of Irregularity: Ax (xex)
3) Axiom of Uniqueness: EQ,Ax (Qex)
4) Axiom of Comprehension: Ex,Ay:y=/=Q (yex<->P(y))

This theory allow unrestricted comprehension without leading to
Russell's paradox.

From this set theory, some of what is regarded as axioms in ZFC, with

slight modefication would be theorums here.

Theorum 1. ExAy (yex)
Theorum 2) ~ExAy ~(yex)
Theorum 3) Pairing : as in ZFC
Theorum 4) Union: as in ZFC
Theorum 5) Separation: AyExAz:z=/=Q ( zex <-> zey /\ P(y) ).
Theorum 6) Replacement: AxE!y:P(x,y) -> AaEbAy:y=/=Q,yeb<->Exea:P(x,y)

Theorum 7) Power: as in ZFC
Theorum 8) Choice: as in ZFC
Theorum 9) Infinity: EN:QeN /\(Ax:xeN -> {m|mex v m=x' }eN )
x' is the complementary set of x , i.e. x':Az:z=/=Q (zex' <-> z!ex)
N= the set of all ordinals


According to 9) the set of all ordinals exist. I think the cardinal of
all cardinals would also be
a theorum in this set theory.


In this set theory it can be proved that for some sets, they are
identical as their power sets.
Also Cantor's proof of non bijectability of x to P(x) is not valid
here.

Is there anybody think that this theory is inconsistent,or incomplete.


Zuhair

Zuhair,

Those are some interesting notions.

I've written about all sets being irregular before, in the last year or
so. Basically I've been conserving that notion. I found that I would
prefer all the infinite sets being irregular, with the finite sets
being regular. Otherwise basically everything's an infinite set,
irregular.

I don't why I have a vague sense that infinities are double irregular,
i.e. they
have more irregularity than the finites, of course I am talking about
this theory
were all sets in it are irregular. Anyhow


Now, there's not so much wrong with that, where the
smallest speck of dust is an integral part of the entire universe, for
real platonists. That leads to consideration of the
dually-self-intraconsistent ur-element of the null axiom theory, A
theory.

As far as incompleteness goes, a Goedelian heuristic is that if you can
do arithmetic in the theory, Goedelians then go about saying it's
incomplete. While that may be so, it is said that Presburger
arithmetic, with addition on zero and successors, is complete where
Peano arithmetic is incomplete, with addition and multiplication on
zero and successors, where a finite proof of an identity in one is a
finite proof of an identity in the other and all the identities in one
are identities in the other, in expansion of multiplication to finitely
many additions.

The notion of N being all the ordinals, or ubiquitous naturals besides
ubiquitous ordinals, that's another notion which I haven't heard of
many, that is to say any, bring forward, in a modern sense. You can
read what I have written about them in those terms.

The universe is infinite.

Ross

Where can I read your opinions about this subject?

Zuhair

.

Relevant Pages

  • Re: Z.Irregular Set Theory
    ... Axiom of Uniqueness: EQ,Ax ... Theorum 2) ~ExAy ~ ... According to 9) the set of all ordinals exist. ... prefer all the infinite sets being irregular, ...
    (sci.math)
  • Re: Z.Irregular Set Theory
    ... Axiom of Uniqueness: EQ,Ax ... Theorum 2) ~ExAy ~ ... prefer all the infinite sets being irregular, ... finite proof of an identity in the other and all the identities in one ...
    (sci.math)