[FOM] Extrapolation Principle

[Zuhair Abdul Ghafoor Al-Johar]

Dear Sirs, 

In relation to earlier postings to this list regarding trials to generalize 
from the hereditarily finite realm of sets to the whole realm of sets (Links 
supplied as postscripts below), the below link exposit a principle that is 
somewhat simple, termed here as the "Extrapolation Principle", it informally
mounts to saying that every scheme having a schematic expression of the form:

For all x1..xn Exist x For all y (y in x <-> phi) 

where phi has no more than two occurrences of atomic expressions in it, where 
an atomic expression is taken to mean an atomic formula of the set language or 
a metatheoretic expression of the form Q(y,x1,..,xn) or of the form F!(y,x1,..,xn),
the first ranging over all formulas of the set language having y,x1,..,xn as
their sole free variables, the second only ranging over all formulas of the set
language that define functions; if that scheme hold true when all quantifiers in
its sentences are bound by the set of all hereditarily finite sets, then it holds
true over the whole domain of discourse. 

If we upgrade this principle as to allow three atomic expressions in phi, 
then we get all axioms of ZF-Regularity. 

Details at: 
https://sites.google.com/site/zuhairaljohar/extrapolation-principle 

Best regards, 

Zuhair Al-Johar 
PS: Links to earlier attempts at: 
https://www.cs.nyu.edu/pipermail/fom/2015-November/019354.html 
http://www.cs.nyu.edu/pipermail/fom/2011-December/016055.html 
http://www.cs.nyu.edu/pipermail/fom/2011-December/016062.html