# [FOM] Size theory.

Zuhair Abdul Ghafoor Al-Johar zaljohar at yahoo.com
Wed Apr 16 22:05:04 EDT 2008

```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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