Re: Size Theory.
- From: Butch Malahide <fred.galvin@xxxxxxxxx>
- Date: Sun, 30 Mar 2008 23:59:57 -0700 (PDT)
On Mar 30, 11:55 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On Mar 30, 7:14 pm, Butch Malahide <fred.gal...@xxxxxxxxx> wrote:
On Mar 29, 10:02 am, Zaljo...@xxxxxxxxx wrote:
[. . .]
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:
[. . .]
Axiom1: Z is standard infinite & Z is Dedekindian finite
[. . .]
Axiom4: Sa<Sb -> ~Sb<Sa
[. . .]
Axiom6:[Ef(f:a->b,f is injective) &
Af((f:a->b,f is injective) -> ~ f is surjective)] -> Sa<Sb
[. . .]
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'.
[. . .]
I will derive a contradiction from your Axioms 1, 4, 6, and 8.
LEMMA: There exist a D-finite set d and a sequence of sets y_n such
that d = U{y_n : n in omega} and, for each n in omega, y_n is a subset
of y_{n+1} and a proper subset of d.
PROOF: By Axiom1, Z is a D-finite infinite set. Let S be the set of
all finite subsets of Z.
CASE I: S is D-finite.
Let d = S and let y_n be the set of all subsets of Z having at most n
elements.
CASE II: S is D-infinite.
Then there is an infinite sequence s_0, s_1, s_2, ... of distinct
elements of S. Let y_n = U{s_0,...,s_n} and let d = U{y_n : n in
omega}.
How do I know d is D-finite in this case?
Because d is a subset of Z.
Let x = {y_n : n in omega}, and let c be a set obtained by removing
one element from d. It follows from Axiom6 that Sy < Sc for each y in
x.
Don't see this either.
Recall that d is a D-finite infinite set, c = d-{i} for some i in d,
and y is a proper subset of d; moreover, it's easy to see that d-y is
infinite. Choose an element i' in d-y, and let c' = d-{i'}. Since y is
a proper subset of the D-finite set c', it follows that there is an
injection from y to c', but there is no bijection from y to c'. Since
there is a bijection between c and c', it follows that there is an
injection but no bijection from y to c. Now Sy < Sc follows by Axiom6.
.
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