Size Theory. corrected again

This is the last version of Size theory:

Size Theory is set of all sentences entailed ( from first order logic
with identity '=' and epsilon membership 'e' and the primitive
constant Z , and the primitve two place relation symbole '< ' to
denote 'smaller than' ,and the primitive one place function symbole
'S' to denote 'size') by the non
logical axioms of ZF and the following non logical axioms:


Define: Sa>Sb <->Sb<Sa

Define: x is standard finite <-> ER (R is a well ordering on x
and converse(R) is a well ordering on x)

Define: x is standard infinite <-> ~ x is finite


Axiom1: Z is standard infinite & Z is Dedekindian finite
Axiom2: Sa=Sb <-> Ef(f:a->b, f is bijective)
Axiom3: Sa<Sb -> ~Sb<Sa
Axiom4: Sa<Sb<Sc -> Sa<Sc

Define: a comparable_to b <->
(Ef(f:a->b,f is injective) or Eg(g:b->a,g is injective))


Axiom5: a comparable_to b ->
[Sa<Sb <->(Ef(f:a->b,f is injective) & Af((f:a->b,f is injective) ->
~ f is surjective))]


Define: x is nested <->
Ayz ((yex&zex) -> (y subset_of z or z subset_of y))

Axiom6: SZ=Z

Axiom7: [x is ordinal &
Ay ((y is ordinal & Sy=Sx)->x in y)]->Sx=x


Axiom8: ~ c comparable_to Ux ->
[(Ay(yex-> Sy<Sc) & x is nested & c is infinite dedekindian finite &
Ux is dedekindian infinite) -> SUx <= Sc]


were '<= ' denote 'smaller than or equal'.


/ Theory definition finished.

There is another version of axiom 8 but perhaps its weaker.

Axiom8:

(c is infinite dedekindian finite & x is dedekindian infinite & ~ c
comparable to x) ->
[Ay((y subset_of x & y is infinite dedekindian finite)->Sy<Sc)
->Sx<Sc]

I was contemplating adding another two axioms to this theory:

Axiom of Para-continuity:


Ax (x is dedekindian infinite -> ~Ey (Sx<Sy<SPx))


Define: Sy>>Sx <-> ~Em (Sx<Sm<Sy)


Axiom para-comparability:


Axyzu ((Sy>>Sx & Sz>>Sx & Su>Sz) -> Su>Sy)

One set theoriest say that the theory is VERY likely to be consistent
and he thinks it is a quite interesting idea, but he is wondering
weather there are mathematical consequences of interest.


Zuhair




.

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