[FOM] The Finite Principle

[Zuhair Abdul Ghafoor Al-Johar]

Dear FoMers,
This topic comes as a continuation to topics posted
to this forum. The last version of Generalization
axiom scheme (Link provided below) also turns to be
inconsistent when added to the scheme of impredicative
class comprehension. Here I shall present what I think
is a simpler and a more convincing version of this scheme
that I also think it would turn to be safer.

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(Notation found at the end of this account):

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.

/

Best Regards

Zuhair Al-Johar

PS: Link to Generalization axiom scheme
http://www.cs.nyu.edu/pipermail/fom/2011-November/015965.html
This turned to be inconsistent with impredicative class comprehension
because the formula
[k](k equinumerous to z1 & k hereditarily strictly subnumerous to y
& y=Uk U k & y is an ordinal)
would define a set that is an element of a set for every z1 that is
an element of a set. but when z1 is Omega_0 it would lead to
Burali-Forti paradox.
Notation found at: http://zaljohar.tripod.com/logic.txt