[FOM] Size theory.

[Zuhair Abdul Ghafoor Al-Johar]

Hi all,

Anybody can point an inconsistency with this 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=y)->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) 

Axiom of non comparability:

Ax (Sx=x -> SPx=Px)

were 'Px' refers to 'power of x'.


***************

Now according to this theory we can have a Dedekindian
finite set Z that has bigger size than any aleph_x.

And the addition, multiplication and exponentiation
rules of algebra of finites will be applicable on Z.

More over we have 2Z > Z+w
were w refers to w

While on the other hand we don't have Z-w.

This theory manage to compare between sizes of
dedekindian finite infinites on one hand and
dedekindian infinites on the other hand, something
that traditional set theory cannot manage to do.
*************
Closary:

A : universal quantifier
E : existential quantifier
~ : Negation
& : conjunction
or : disjunction
-> : implication (uniconditional)
<-> : biconditional

Zuhair





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