Re: Z.Ulternative Set Theory


smn wrote:
zuhair wrote:
smn wrote:
zuhair wrote:
Hi,

At last I have finished from building this set theory.

- Ulternative Set Theory-


Primitive: e


6) Separation



Ax:10) Axiom of Universe


ExAy yex .




Is their anyone who think that this theory is inconsistent?
Yes.6) and 10) are incompatible using the standard Russell argument -

Let U be a universe .(Ax 10) so Ax(xeU)
6) if I understand this instance separation properly : AyEx (zex
<--->zey & z!ez) .Take y=U

then using 10) (ie xeU always holds) we get : Ex (zex<-->zeU & z!ez)
<--> Ex(zex<-->z!ez)

Let R (for Russell) be such an x .Then zeR <---> z!ez .Then as usual
take z=R to get

ReR <--->R!eR which is a contradiction.

Yes, I know this very well. But the contradiction is in membership of
U, it is not in the theorums which will spring from this theory. My
thinking is that contrdictive sets do exist as sets, they pose no
problem whatsoever on theoratic processing of this theory.

But you are right regarding dropping the axiom of separation, since
there is the axiom of replacement, then there is not need for
separation.

Either drop axiom 10) or call
your objects (what your variables denote (x,y,z,....) classes and
define x is a set iff Ey (xey) and then revise

10) to 10)' : Ex Ay (yex <--> y is a set) and call this x by U =x (the
universe of sets) .This is the Kelley -Moore system which is stronger
then Zermelo -Fraenkel .The Rest having to do with allowing xex and
friends is a denying of the regularity axiom which you are friee to do
with know known contradictions.Regards smn

In reality I have tried something like this in another thread, but yet
I will try to put it here.

So the theory would appear like that:


-Z.Ulternative Set Theory-

Primitive e
Definition: x is a set <-> Ey (xey)

Axioms:
1) Extentiality 2) Universe: Ex Ay ( yex <-> y is a set ) 3) Empty set
4) Circularity 5) Pairing6) Complementary set 7) Irregularity
8) Ambiguity 9)Union 10) Infinity 11) Intersection 12) Harmony
13) Synchrony 14) Replacement 15) Power 16) Choice.

were Irregularity is what I called the axiom of irregular complementary
set.

Zuhair

Fine,now lets pretend we have recovered separation from
replacement.Lets also name the universe (unique by extensionality ) "U"
Let R be the class (objects denoted by variables) obtained from
separation by requiring xeR <--->xeU and x!ex <--->xis a set and x!ex.

Replacing x by R gives ReR <---> R is a set and R!eR .From this it
follows that R is not a set. Non sets are called proper classes.Since R
is a subclass of U it follows from the power set axiom that U is not a
set either. However Ax( x is a subclass of U) .So all your objects are
subclasses of the universe although some are not members of U.

.You probably know all this but I wanted to check.

Yea, I know this of course.

This gets rid of contradictive sets.I strongly disagree that they cause
no difficulty.Its true that in real life (serious ) reasoning using
ordinary language there are contradictions that one avoids by watching
out that you are making sense or using empirical checks using the
senses ( eyes ,ears..) but in mathematics with its much limited scope
but where the arguments get much more complicated it is intolerable to
have contradictions in the foundation.Regards,smn

Accordingly, this theory is consistent!

Zuhair

.

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