Re: For All x
- From: hurburble@xxxxxxx
- Date: Fri, 20 Mar 2009 18:32:59 -0700 (PDT)
On Feb 28, 7:34 pm, apoorv <sudhir...@xxxxxxxxxxx> wrote:
On Feb 12, 1:00 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
Re Feb 10, 12:02 pm, MoeBlee <jazzm...@xxxxxxxxxxx>:
P.S. If one doesn't learn set theory from books and/or organized
lectures, then just how does one learn the subject? By flitting from
one Wikipedia page to another, with no overall conceptual and
technical context among them, complemented by various ad hoc
conversations with experts, sincere amateurs, and outright
ignoramuses, and then supplemented by watching one's favorite episodes
of "Friends"?
MoeBlee
Well, I am only saying that we need to be alive to other perspectives.
(after all, for 1500 years, the sun went round the earth) In any model
of ZF set theory, the variables range over a domain D specified
outside the theory. The domain D is itself a set, yet its existence is
postulated outside the theory. Further, when we say, for example,
Ax ~x e x, we are actually saying, Ax in D, ~x e x.
So, we actually are using D as a constant in the theory ( is that
permissible always?), and two different relations
Namely, “in” between objects and domain, and “e” between objects.
If we are looking for a self contained set theory, the existence of
the domain D
must be postulated within the theory.
Consider a theory with these axioms:
1)E z Ax x e z (existence of domain D)
2) E z Ax ~ x e z (existence of null set 0)
3)Ax E z Ay ( ye z <-> y e x or y=x) (existence of successor Sx of
x)
Axioms 2 &3) together imply that D is infinite and a separate Axiom
of infinity is not
needed. Because of 1) D contains itself. We consider a model with the
smallest possible domain D. D would be the
set D=[0,1,2 . . .D}
Having postulated the existence of the domain D, we need to give up
full fledged separation. In particular, we can see that
~E y Ax [x ey <->x e D and ~x =D}, that is, the set D-{D} =
{0,1,2 . . .} does not
exist.
In this model,
a) There is only one infinite set, which is its own successor ..
b) Infinity cannot be separated from itself.
If “e” is interpreted as ‘less or equal’ and Sx as any function ,
then every sequence
In D is a model for the theory and D is the limit for that sequence..
So, we have a choice , we either accept full fledged separation, and a
unending escalator
Into the transfinite, or restrict separation and work with a single
infinite domain.
-apoorv
But it is impossible to "postulate the existence" of any D of a theory
within the theory itself. If it is something to be done, it is to be
done only with inconsistent theories!
.
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