# [FOM] The Cumulative Hereditary Hierarchy.

Zuhair Abdul Ghafoor Al-Johar zaljohar at yahoo.com
Wed Jan 4 00:10:57 EST 2012

```Dear Sirs:

A closely related question is: what would be the strength of
the following first order theory with the following axioms?:

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

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

1.Construction: if phi is a formula in which x is not free,
then ([!x](y)(y e x <-> elm(y) & phi)) is an axiom.

2.Pairing: (A)(B)(x)((y)(y e x -> y=A ? y=B) -> elm(x))

3.Hierarchy: (i)(x)(i is ordinal & elm(i) & x c H_i -> elm(x))

where for every ordinals i,j:

H_0 = 0
H_i+1 = {x| x hereditarily subnumerous to H_i}
H_i = U(H_j)j<i, when i is a limit ordinal.

/

My guess is that this theory can prove the existence of any
element Von Neumann ordinal a as long as a < Omega_a. So I
think this theory is much weaker than ZF.

Best Regards

Zuhair Al-Johar

PS:Notation
(x): for all x
[x]: there exist x
[!x]: there exist unique x
?: disjunction
c: subset of
:= stands for "is defined as".

On Sun, 1 Jan 2012 09:01:00 -0800 (PST)
Zuhair Abdul Ghafoor Al-Johar <zaljohar at yahoo.com> wrote:

> Dear FOMers,
>
> Define the Cumulative Hereditary Hierarchy H as the union
> of
> all the following stages:
>
> H_0=0
> H_i+1=H(H_i) for any Von Neumann ordinal i
> H_i= U(H_j) j<i, for any limit Von Neumann ordinal i
>
> where H(x) is the set of all sets hereditarily subnumerous
> to x.
>
> Questions:
>
> 1) Can ZF alone (i.e. without Choice) prove the existence
> of H_i for
> each ordinal i ?
>
> 2) Is the cumulative hierarchy V of Von Neumann's a
> subclass of H ?
>
> 3) If 2) is true, then can there exist a set in H that is
> not equinumerous
> to any set in V.
>
> Happy new year!
>
> Zuhair
>
>
```