Re: Size Theory.
- From: Rupert <rupertmccallum@xxxxxxxxx>
- Date: Sat, 29 Mar 2008 21:23:57 -0700 (PDT)
On Mar 30, 11:03 am, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On Mar 29, 11:02 pm, Zaljo...@xxxxxxxxx wrote:
Hi all,
In a previous post titled 'Zuhair's numbers', I introduced the idea of
comparability between infinite sets that are Dedekindian finite
and Dedekindian infinite sets.
I will present the newer version of it here.
Size Theory is set of all sentences entailed ( from first order logic
with identity 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
Axiom1: Z is standard infinite & Z is Dedekindian finite
Axiom2: (x is Dedekindian infinite & Sx<SZ) ->
~Ey (y subset_of x & y is standard infinite & y is Dedekindian
finite)
Axiom3: Sa=Sb <-> Ef(f:a->b, f is bijective)
Axiom4: Sa<Sb -> ~Sb<Sa
Axiom5: Sa<Sb<Sc -> Sa<Sc
Axiom6:[Ef(f:a->b,f is injective) &
Af((f:a->b,f is injective) -> ~ f is surjective)] -> Sa<Sb
Define: a comparable_to b <->
(Ef(f:a->b,f is injective) or Eg(g:b->a,g is injective))
Axiom7: (a comparable_to b & Sa<Sb) ->[Ef(f:a->b,f is injective) &
Af((f:a->b,f is injective) -> ~ f is surjective)]
Axiom8:AcAx ([Ay (yex-> Sy<Sc) &
Ayz ((yex&zex&~y=z) -> (y proper subset_of z or z proper subset_of
y))] -> SUx <= Sc)
were '<= ' denote 'smaller than or equal'.
/ Theory definition finished.
Now according to this theory we have Omega < Z
because every member of Omega is smaller than Z,
and Omega is the union of its members
and since every two different members of omega x, y
we have x proper subset of y or y proper subset of x
then by axiom 8 , Omega <=Z
Now from axiom 3 Omega=Z iff there exist a bijection between them, but
it is a theorem of ZF that there do not exist such a bijection, thus
Omega < Z.
The same will apply to Omega +1 , Omega +2,...., Omega^2,...
also same will apply for Omage_1,Omega_2 ,etc....
all would be smaller than Z.
Now Z itself is smaller than Z+1 < Z+2 < Z+3,.........
also we have Z-1 > Z-2 > Z-3,............
the union of Z+n for n=1,2,3,..... is Dedekindian infinite
and it is Z+w and this will be strictly smaller than 2Z,
Be careful. How do you prove that the disjoint union of Z and omega
has smaller size than the disjoint union of two copies of Z?
Sorry. I now see how you do this. This means that my attempted
interpretation of your theory in ZF was wrong again. However I now
think that I can do a consistency proof for your theory (relative to
ZF) by transfinite recursion. As far as I can tell, my other comments
still stand.
.
- References:
- Size Theory.
- From: Zaljohar
- Size Theory.
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