The Finite principle: if Φ(y) is a formula such that
when all its parameters stand for hereditarily finite "HF"
sets then it follows that there exist a set {y|Φ(y)}that
is hereditarily finite, then there would exist a set
{y|Φ(y)}that is an element of a set as long as all
parameters of Φ(y) stand for elements of sets.

In symbols:

Def.) elm(x):= [y](x e y)

elm(x) is read as: x is an element.

The Finite Principle:
if Φ(y) is a formula in which x is not free and
where z1,...,zn are all its parameters, then

(z1)...(zn)(HF(z1)&...&HF(zn)->[x](x={y|Φ(y)} & HF(x)))
->
(z1)...(zn)(elm(z1)&...&elm(zn)->[x](x={y|Φ(y)} & elm(x)))

is an axiom.

Now this axiom scheme coupled with the following
axioms would prove ZF and Con(ZF)

Construction: if Φ is a formula in which x is not free,
then ([!x](x={y|elm(y)&Φ})) is an axiom

Infinity: (x)((y)(y e x -> HF(y)) -> elm(x))

HF which stands for the predicate
"Hereditarily Finite" is defined as
a finite set where every element of
its transitive closure is also finite.

The transitive closure of a set is defined
as the minimal transitive superset of that
set.

Finite set is defined as a set having maximally
two elements or a set that is bijective to
a subset of a natural number.

A natural number is defined as an empty set
or a successor Von Neumann ordinal in which
every element of it is either empty or is
a successor Von Neumann ordinal.

A Von Neumann ordinal is defined as
a transitive set of transitive sets in which
every subset of it have a disjoint element
of it.

Of course every object in this theory is
to be termed as a set, a set that is
not an element of a set is to be termed as
a proper set. This will obviate the need
for class terminology.

/

This theory offers some kind of natural explanation
to ZF, seeing that ZF is provable from a theory
that basically pivots around three themes:
that of "Construction" of sets, and that of
"Mimicking the Finite world of sets" and
that of a desire to have "Infinite sets"; is
really a nice explanation, this somehow
motivates ZF, since after all it is constructing
sets by mimicking hereditary finite set construction
and this seems to be reasonable intuitively. 


Zuhair Al-Johar
15/12/2011

PS: Notation at:http://zaljohar.tripod.com/logic.txt