Size notion is a hextuple (V,W,S,=,<,>) where V,W are sets
and S, denoting Size, is a one place function whose domain is V and range is W, = denoting equality, < denoting strict smaller > denoting strict greater than, the last three are binary relations on W; that fulfills the following criteria:
1. S(x)=S(x)
2. S(x)=S(y) -> S(y)=S(x)
3. S(x)=S(y) ^ S(y)=S(z) -> S(x)=S(z)
4. ~ S(x)<S(x)
5. S(x)<S(y) -> ~S(y)<S(x)
6. S(x)<S(y) ^ S(y)<S(z) -> S(x)<S(z)
7. S(x)<S(y) <-> S(y)>S(x)
8. S(x)=S(y) ^ S(y)<S(z) -> S(x)<S(z)
9. S(x)=S(y) ^ S(z)<S(y) -> S(z)<S(x)
10. S(x)=<S(y) -> exists f (f:x->y, f is injective)
11. S(x)=S(y) -> exist f (f:x->y, f is bijective)
12. exist f (f:x->y, f is injective) ^ ~exist g (g:y->x, g is injective) -> S(x)<S(y)
Those are the least demanded characteristics of a size notion.
Any other characteristics added would serve to further restrict (in some sense or another) this size notion, I'll present two known ones:
13. exist f (f:x->y, f is injective) -> S(x)=<S(y) [Cantor]
14. x proper subset of y -> S(x)<S(y)
Now a size notion that fulfills all of the above 14 criteria would only suit Dedekind finite sets, so it has a very restricted application.
Cantor's Cardinality fulfills the first 13 criteria.
However we can dispense with 13 and keep 14 and add further criteria as to develop an alternative size notion to Cardinality. It would be a size notion since it fulfills the basic 12 criteria of a size notion.
Here I'll re-post those additional criteria of this alternative notion, Here it will be restricted to sets of natural numbers only, i.e. subsets of N.
Define: P is n-equi-partition on N iff n in N ^ P={[nj, nj+n) /\ N | j in N}
Define: A is an n-choice set on P iff A is a subset of the union of P and A has exactly n elements of each element of P as elements of it.
Define: f is a choice preserving function iff f is a bijection from A to B both being sets of naturals that are m-choice sets on an n-equi-partition P on N, for some m,n where m=<n, such that for every x in A, for every k in P: x in k <-> f(x) in k.
Definitional criteria: for sets A,B of natural numbers:
15. S(A)=S(B) <-> exist f (f is a subset of a choice preserving function ^ A=dom(f) ^ B=rng(f))
16. S(A) < S(B) <-> exist C (C proper subset_of B ^ S(A)=S(C))
(of course criterion 13 is omitted).
Now this notion of size (satisfying all above criteria except 13) would work fine on sets of naturals (I think?!) It might be possible to extend it to sets of ordinals, but
it would need some modification to define equi-partitions in such a manner as to avoid conflicts.
Cardinality is more general in the sense that it can decide on size comparisons between any two sets as long as choice or foundation is held. On the other hand the alternative notion is not that general, there is wide area of incomparability left with it.
A nice issue raised is about uncountability of the alternative sizes of sets of naturals? which might be the case. I don't know the real answer to it.
The above approach came historically first, the following (May 26 2017) is possibly a nicer one:
Definitions:
A utterly disjoint B iff
for all a in A, for all b in B (a<b) or for all a in A, for all b in B (b<a)
P is a nice partition on N iff
U(P)=N ^
for all x in P Exist m (m in x) ^
Exist a,b in P (~a=b) ^
for all c,d in P (~c=d -> c utterly disjoint d)
So a nice partition on N is a set of many non empty sets that are pairwise utterly disjoint, the union of which is N.
X is a choice set on P iff X subset U(P) ^ for all c in P exists x (x in X ^ x in c)
f is a choice preserving function iff f is a bijection from A to B both being choice sets on a nice partition P on N such that for every x in A, for every c in P: x in c <-> f(x) in c.
If this is consistent with criteria 1 to 16 except 13, then it is to be the one adopted since it subsumes the above one, and so restrictions would be redundant, besides that it might be stronger.
I'd conjecture that this method can be extended as to include any set of ordinals, but we need to further restrict all compartments in nice partitions to be "finite".
Zuhair