[FOM] Cardinality Beyond Regularity and Choice!

[Zuhair Abdul Ghafoor Al-Johar]

Hi all, 

We may say that a Cardinal is a function from classes to classes such that for any classes A and B: 

Cardinality (A) = Cardinality (B)  iff  A equinumerous to B. 

Were A equinumerous to B iff 
(there exist an injection from A to B and there exist an injection 
from B to A) 

Also it is desirable to have the following property added to the 
above: 

For all A: A is a set -> Cardinality (A) is a set. 

Cardinality might be stipulated as a primitive concept, i.e. 
a primitive one place functions symbol "| |" added to the list of 
primitives of the language of ZF\NBG\MK, and axiomatized with the 
above two assumptions: 

1. For all x,y ( |x|=|y| <-> x equinumerous to y ) 
2. For all x ( x is a set -> |x| is a set ) 

Of course the second condition is for theories permitting the 
existence of "proper" classes in their universe of discourse, like 
NBG\MK\Ackermanns'(e,=,| |),etc.., 
for ZF(e,=,| |) only 1 is axiomatized. 

This primitive Cardinality is the most general approach to 
Cardinality. 

The second approach is to *DEFINE* cardinality as special kind of sets in such a manner as to satisfy the above two conditions. 

It is well known that Cardinals are defined under the assumptions of Choice or Regularity. 

The question that present itself is: 

Can we define Cardinality for every set in ZF minus Regularity? 
i.e. beyond Choice and Regularity. 

Lets examine, the cardinals that we know 

(1) Frege-Russell's Cardinals: 

Cardinality(A) is the class of all sets equinumerous to A. 

      Those are incompatible with Z, since 
      they entail the existence of the set of all sets in Z, or in 
      NBG\MK they would be proper classes. However in 
      NF and related systems they are as general as the primitive 
      concept of Cardinality, but the problem with these theories is 
      that they are very complex, and difficult to understand, using 
      concepts of stratification of formulas which is not desirable, 
      even the finite axiomatization of NFU , though its axioms 
      do not use stratification, but yet most of its theorems 
      relies on it. 

      Those Cardinals were the first defined cardinals 
      in history of human kind. 

(2) Von Neumann's Cardinals. 

   Cardinality(A) is the least of all Von Neumann 
   ordinals equinumerous to A. 

Those depend solely on Choice, so they cannot survive beyond it. 

(3) Modified Scott Cardinals: 

These are defined for every set A as follows: 

Cardinality(A) is the set of all well-founded sets equinumerous to 
                    A of the least rank. 

Scott Cardinals do not require Choice, but they require 
Regularity,however the above amended definition works in 
absence of Regularity, but the problem is that the sentence 
" every set is equinumerous to a well founded set" (Coret's axiom) is not a theorem of ZF, so it must be axiomatized. 

So these Cardinals work in ZF-Regularity+ Coret 

So to some extent they do work beyond Regularity and Choice. 

(4)My version of Cardinals: 

Those are the Cardinals I defined as: 

Cardinality(A) is the class of all sets equinumerous to A having every member of their transitive closures strictly subnumerous to A. 

Or simply: 

Cardinality(A) is the class of all sets equinumerous to A that are 
hereditarily strictly subnumerous to A. 

In ZFC those cardinals are non empty sets. However those Cardinals unlike Von Neumann's do not necessarily require full choice, they actually require the following assumption. 

(Z)For every set x: Cardinality(x) is a non empty set 

or in ZF style 

(Z)For every x Exist y ( y=Cardinality(x) & ~y=0 ). 

Now that assumption if added to ZF as an axiom. 
then ZF+Z is weaker than ZFC, perhaps strictly weaker actually. 

Actually we can drop Regularity as well, so we can have 
ZF+Z-Regularity. 

Now this definition works in ZF+Z-Regularity. 

So these cardinals do work beyond Regularity and Choice, provided the assumption Z above. 

The interesting matter is to see if ZF+Z-Reg would be 
weaker or stronger than ZF-Reg+Coret, or perhaps 
neither weaker nor stronger! they might simply work 
under different conditions beyond Regularity and Choice. 

The interesting thing in the last definition of mine, is that it 
doesn't require Coret's axiom, which is a step downward in diving 
beyond Regularity and Choice. 

Other definitions: Actually all of them are nothing but versions of my definition of Cardinality

(1)Cardinality (A) is the class of all sets equinumerous to A that are hereditarily subnumerous to A. 

(2)Cardinality(A) is the class of all hereditarily hereditary sets equinumerous to A.

Were a hereditary set is defined as a set that is strictly supernumerous to every member of its transitive closure.

Hereditarily hereditary set is a set that is hereditary were every member of its transitive closure is hereditary.

(3)For any set A, if there exist a set B that is equinumerous to A
such that there do not exist a set C that is equinumerous to A 
and TC(C) strictly subnumerous to TC(B), then B is said to have a 
minimal transitive closure, and 

The class of all sets equinumerous to A with transitive closures 
equinumerous to TC(B) is said to be the Cardinality of A 

Or more generally: 

A Cardinal is an equivalence class of sets of minimal transitive 
closures, under equivalence relation 'bijection'. 

so:

Cardinality(A) is the class of all sets equinumerous to A, having 
minimal transitive closures.

All the above versions of Cardinality can be axiomatized to be non empty sets like the original version of My definition of Cardinality, and work in conditions weaker than choice and regularity.

During my approach to justify such approaches, I defined 
Z+Size limitation theories as below:

'Z+Size Limitation' is the set of all sentences entailed (from First 
Order Logic with Identity ' =', and Epsilon membership 'e' ) by the 
following non logical axioms: 

1) Extensionality: for all z ( z e x <-> z e y ) -> x=y 

2) Regularity: 

For all x ( Exist y (y e x) -> 
               Exist z ( z e x & ~ Exist c (c e z & c e x))) 

3) Separation: if phi(y) is a formula in which at least y is free, 
and in which x is not free then all closures of 

For all c Exist x for all y ( y e x <-> ( y e c & phi(y) ) ) 

are axioms. 

4) Pairing: For all z,y Exist x ( z e x & y e x ) 

5) Union: For all c Exist x for all y,z ((z e y & y e c) -> z e x) 

6) Power: For all c Exist x for all y 
                (for all z (z e y -> z e c) -> y e x) 

7) Infinity: Exist N (0 e N & for all x (x e N -> xUnion{x} e N)). 

8) Size Limitation 1:If phi(y) is a formula in which at least y is 
    free, and in which x is not free, then all closures of 

~ for every ordinal d Exist x ( for all y (y e x -> phi(y)) & 
                                d equinumerous to x ) 

<-> Exist x for all y ( y e x <-> phi(y) ) 

are axioms. 


9)Size Limitation 2: 

If phi(y) is a formula in which at least y is free, 
and in which x is not free, then all closures of 

Exist s for all x,z ((for all y ((y e x -> phi(y)) &  z e TC(x)) 
                            -> z strictly subnumerous to s) 

-> Exist x for all y ( y e x iff phi(y) ) 

are axioms. 

were "equinumerous"  and "strictly subnumerous" are defined in the standard manner.

TC stands for "transitive closure" as defined in ZF (replacement is a theorem of Z+SL1).

Theory definition finished/

This theory that I called Z+SL1,2 do prove the existence of the set of all sets hereditarily subnumerous to x , for every x actually, lets denoted as H(x)

However it doesn't prove that for every x, x subnumerous to H(x).

Thus this should be axiomatized, to have the versions of Cardinals work inside such theories.

Or we can axiomatize the following:

For all x Exist y 
(y equinumerous to x  and
not Exist z (z equinumerous to x and TC(z) strictly subnumerous to TC(y)))


>From all the above, it is seen that we can indeed have defined Cardinals working under different conditions beyond Regularity and Choice, however till now I don't know of a defined Cardinal in ZF minus Regularity, without having any additional assumptions.

One of the additional assumptions that I thought would be worth considering is a strong version of Extensionality, were any
two sets having the same "proper" members are identical, were proper members refer to members of a set other than the set itself, so in any theory in FOL with identity we add the following axiom instead of Extensionality:

For all x,y (For all z ((z e x & ~ z=x) iff (z e y & ~ z=y)) -> x=y)

This would shun Quine atoms from existence, and shun proper classes of them from existence. Since these atoms can form sets that are not well founded and not equinumerous to a well founded set, so I thought that Scott Cardinals would work in ZF-Reg having strong axiom of Extensionality instead of the ordinary one, so I thought that this will be enough and we will no longer need Coret's axiom for those cardinals to work in a general manner. However this was nothing more than a guess.


The interesting question after all of that is:

Seeing that it is very difficult to define Cardinality beyond Choice and Regularity, and seeing how complex the definitions become when we try to do so, then Why not just adopt the "primitive" approach to Cardinality, it seems both simpler and most general. The primitive approach can even define Cardinality of proper classes, something that no defined Cardinal can achieve Frege-Russell's cardinals inclusive, in these circumstances it seems that it would be provable that the cardinality of any proper class would be a proper class, otherwise we will end up with the class of all sets being strictly supernumerous to itself, which is a contradiction.

So to me I see primitivizing Cardinality as a reasonable(or possibly superior) alternative approach to defining it.

Wouldn't it be?

Zuhair