Re: Cantor and the binary tree
- From: mueckenh@xxxxxxxxxxxxxxxxx
- Date: 3 Jul 2005 08:54:35 -0700
Virgil wrote:
> > The axiom of infinity creates both, numbers and sets. Show why,
> > nevertheless, according to your opinion, infinite sets but no infinite
> > numbers are created although this axiom works in the same way in both
> > cases.
>
> Because, except in it name, that axiom does not mention or use either
> the quality of being finite or of being infinite.
Interesting new insights. So the name is misleadimg, even deceiving?
>
> The axiom, in one form, say merely that there exists sets S such that {}
> is a member of S and for every object x which is a member of S, the
> object (x union {x}) must also be member of S.
>
> It is trivial that the intersection of all such sets is such a set,
Oh, it is equally trivial that the union of all initial segments is an
initial segment.
>and
> that intersection is taken to be the set of naturals, N, though nowdays
> the first member, {}, is taken to be 0 instead of 1 as it usually was
> when I was young.
It "is taken", including nought? But what is the intersection really?
What would follow, if politicians decided that all numbers > 1000 did
not belong to N?
> The quality of a set being finite or infinite is only defined later
It is not a matter of the axioms? Perhaps infinity does not exist at
all after all?
> and
> in terms of N as being s small as a set cold get and still be infinite.
> And the objects in that intersection, by reason of that definition of
> infiniteness of sets are finite sets.
No. Having pressed you to come up with such answers is an important
achievement. One has to put the right questions only, I see. But you
are wrong.
The condition a u {a} e A guarantees the infinity of A if A is taken to
be a set and also if it is taken to be the variable of numerical values
n.
It is very cheap to find a bijection between initial segments
{1,2,3...,n} and numerical values n. You are now in the situation to
defend the position; n is always finite but the union of all segments
{1,2,3...,n} created or guaranteed by the axiom of infinity becomes
infinite somewhere. It is simply silly.
Regards, WM
PS. I will leave this thread because it has become so long that we soon
can distinguish isolated paths. I have started two new threads. You are
invited to contribute.
.
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