[FOM] An explication of extensions, sets and membership.

[Zuhair Abdul Ghafoor Al-Johar]

Dear Sirs, 

I found an inconsistency with the theory at exposition. 

This theory proves that {x} is a set for every x, since if not a set 
then {x} would be empty and thus a sub-extension of any set and thus a set, 
a contradiction. 

Now we can use that to penetrate non sets through parameters of formulas of 
acceptance axioms to get a paradox! take the set {e(Set)}, take the formula 
"Exist y (x in y ^ y in {e(Set)})" which would be a formula in the language 
of set theory, let's call the extension of the predicate defined after this 
formula to be R. Now EVERY element of e(Set) is a set and so would an element 
of R by axiom of acceptance and therefore R would be a set! so by the subsets 
axiom e(Set) must be a set! thus have ALL sets as members of it and clearly 
this is paradoxical, since we can easily recover the Russell set, because R 
would be a set of all sets and the extension of the predicate defined after 
formula "y E R ^  not y E y" would be the Russell's set, a paradox! 

One way to remedy that is to restrict parameters of formulas of acceptance to 
hereditarily set extensions, by then we need to add an axiom of infinity if we 
are to interpret Zermelo. 

A more radical approach is not to grant existence of extensions for predicates 
not definable in the language of set theory, this can be done by weakening axiom 
of rejection as to grant some rejection instances just for the case of unfulfilled 
predicates, formally this is: 

[not Exist x (Q(x))] => Exist y,z (y=<z,Q>) 


This would possibly also allow us to free axioms of acceptance from the restriction 
on parameters altogether. The resulting theory would interpet Zermelo, but I'm not
sure if it can interpret ZF. 

Best Regards, 

Zuhair Al-Johar 




On Tue, 23 Jun 2015 16:42:35 +0000 (UTC), Zuhair Abdul Ghafoor Al-Johar <zaljohar at yahoo.com> 
wrote: 


A blend between Mereology, especially that of David Lewis, 
and Frege's method of extending predicates, would yield a 
body of knowledge that seems to have the potential of aiding 
understanding of classes and sets on the intuitive level as 
well as on the technical side, and so might prove useful to 
motivate further extensions of set theory, investigating 
useful alternatives, and motivating overall research in this 
field. For Details please see:
http://zaljohar.tripod.com/explication.txt