Re: V
- From: zuhair <zaljohar@xxxxxxxxx>
- Date: Fri, 01 Jun 2007 11:51:04 -0700
On Jun 1, 12:32 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Jun 1, 7:57 am, zuhair <zaljo...@xxxxxxxxx> wrote:
On Jun 1, 12:28 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
I can barely make sense any of that. You're presupposing ordinal
arithmetic just to stipulate a certain denumerable set of constants.
I'd suggest considering looking for ways to simplify, since it seems
to be quite a lot if you need ordinal arithmetic just to specify the
symbols of your language.
MoeBlee- Hide quoted text -
- Show quoted text -
To complete previous reply to you in which
I introduced a simpler way of stipulating
a certain denumerable set of constants.
DEFINITION OF OMEGA DIMENSIONAL "oo-ML".
w-dimensional oo-ML: is the set of all sentences entailed (from first
order logic with identity and primitive constants outlined below)
by the following non logical axioms.
Primitives:e,=,for every n=1,2,3,.....
there is a primitive constant V_i1_i2_..._in,
were each ij=1,2,3,..... for j=1,2,...,n-1
and if n>1 then in=2,3,4,........,
and if n=1 then in=1,2,3,4,......
1)Axiom of Extensionality: AxAy(x=y<->Az(zex<->zey)).
2)Axiom of Regularity:Ax(~x=0 -> Ey(yex & y disjoint x)).
3) Comprehension: For every vector I=i2_i3_......_in,if F is a formula
in
which x is not free
then all
closures of
ExeV_((i1)+1)_I Ay(yex<->(yeV_i1_I & F(y)))
are axioms.
Clarification V_((i1)+1)_I = V((i1)+1)_i2_i3_i4_..._in
and V_i1_I=V_i1_i2_i3_....._in
For every vector I=i1_i2_i3_......_in, all the following sentences
are
axioms:
4) Pairing: AreV_I AseV_I ExeV_I Ay(yex<->(y=r v y=s)).
5) Union: AaeV_I ExeV_I Ay(yex<->Ez(zea&yez)).
6) Power:AaeV_I ExeV_I Ay(yex<->Az(zey->zea)).
7) Infinity: ExeV_1( 0ex & Ay(yex->yU{y}ex))
8) Membership: Ax( xeV_I <-> (Ay(yex->yeV_I) & Ez( zeV_I and z
supernumerous_to x))).
with the usual meaning of 'supernumerous_to'.
Strong version of 8) would be:
8) Membership: Ax( xeV_I <-> (Ay(yex->yeV_I) & V_I supernumerous_to
x)).
9) Layers: for every two vectors I=i1_i2_i3_......_in, and
J=i1_i2_i3_........_im, if m<n then
the sentence
Ax(xeV_J -> xeV_I)
is an axiom.
Negation of choice can be added to the theory with weak version of
membership, at each layer, in the following manner.
10) Anti-choice: for every vector I=i1_i2_i3_......_in
the sentence
D_I subnumerous_to V_I
is an axiom
with the usual meaning of 'subnumerous_to'
were x=D_I<-> Ay(yex<->(yeV_I & y is ordinal)).
Of course according to this theory for every vector I
the sentence D_I( ~D_I e V_I) is a theorem.
/
Theory Definition Finished.
Now this theory have a countable number of primitives
that is equal to w+w^2+w^3+...+w^n+w^(n+1)+........
were n=1,2,3,4,.....
So what theory it can be interpreted in
would it be: ZF/ZFC + (w+w^2+w^3+...+w^n+w^(n+1)+........)+1 IC.
and which of the known inaccessible cardinals is bigger than
all cardinals in this theory.
Zuhair
.
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