Re: Another stab at Cantor


Arturo Magidin wrote:
In article <1158691856.458922.182270@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
georgie <geo_cant@xxxxxxxxx> wrote:
Arturo Magidin wrote:
In article <1158683687.456621.233970@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
georgie <geo_cant@xxxxxxxxx> wrote:

Arturo Magidin wrote:
In article <1158683189.567305.51540@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
georgie <geo_cant@xxxxxxxxx> wrote:

Arturo Magidin wrote:

(ii) The strings Dk, where D1 is obtained via the diagonal process
from L1, and D(k+1) is obtained via the diagonal process from the
list that has D(k+1-j) in the j-th position, j=1,...,k, and L1(n) in
the (k+n)th position.

"the list that has D(k+1-j) in the j-th position, j=1,...,k, and L1(n)
in the
(k+n)th position." is an attempt to describe L_Omega.

No, Georgie boy. It is an accurate description of the list considered
in step k+1 of the procedure described. The list described in (ii) is
used ONLY in order to construct the string D(k+1).

Do you use the "No, Georgie boy." proof often?

Are you under the misapprehension that the sentence above was meant as
proof of something? That may be part of your legion of problems.

So did you say it because you can't control yourself?

Have you stopped beating your wife?

I've been doing what you've been doing in this regard.

According to you, (ii) does not describe a list in the limiting case.

(ii), by itself, was never meant to describe the limiting case
proposed by the original poster. (ii) describes one of the two kinds
of elements in the limiting set, namely, the constructed strings. In
order to explain HOW those strings were constructed, since they are
constructed using a list, I described that list. The important thing
about point (ii) is the strings Dk; the rest is a description of how
each string Dk was obtained.

This is the crux, I agree.

As you say:

"(ii), by itself, was never meant to describe the limiting case"
"(ii) describes one of the two kinds of elements in the limiting set"

In summary, (ii) describes numbers that are associated with lists,
but the limiting case is a set.

Namely, the set that contains exactly all strings described in (i) (the
original list), plus all strings described in (ii) (the constructed
strings Dk for integer k). And nothing else.

Since the limiting case is not a list, (ii) must not apply so you
didn't explain why the limiting case can't include all the reals.
I'm not saying it does. I'm pointing out nobody has proven it doesn't.
You can't seem to accept that and have resorted to personal attacks.

L_Omega, as described, is a set indexed by the integers (not a
list). To positive integers correspond the elements of the original
list L1. To nonpositive integer indices correspond the strings
Dk. What are the strings Dk? They are the strings constructed in the
manner decribed in (ii). Point (ii) describes strings that are in the
final set; it is not, except in your febrile imagination, an attempt
at describing the full final set.

I think you are in denial by calling something that isn't a list a
"set indexed by the integers".

And I ->know<- you don't know what you are talking about.

You've yet to impress me with any of your knowledge. But your
personal attacks speak volumes.

Tell me, son. What is it you think a "set indexed by the integers"
means?

If you don't know, then ask. That way you won't make a fool of
yourself.

You brought that terminalogy up. You said the limiting case,
L_Omega, is not a list and then you went on to call it a set
that was indexed by the integers.

A set X is said to be "indexed by the set Y" if there is bijective
set-theoretic map f:Y->X. So saying that the set is "indexed by the
integers" means there is a bijection from the set of all integers
(positive, negative, and zero), to the set in question.

In this case, our set contains exactly the elements L1(k), and the
elements Dn, with k and n positive integers, and nothing else. Call
this set F (for "final").

"our set" is L_Omega? The one you said wasn't a list?

We define the map f: Z --> F by:

if k is positive, f(k) = L1(k).

f(0) = D1

if k is negative, f(k) = D(-k+1).

So: the "index 0" element is D1. The "index -1" element is
D(-(-1)+1)=D2. The "index -2" is D3. The index 100 is L1(100). The
index -85917 is D(85918).

So the set is indexed by the integers.

No, the set as given is not a list, because this set, ->ordered this
way<- (which is the way it is naturally ordered given the procedure
given) is not a list.

However, there is an easy way to make a list which has each and every
element of this set on the list exactly once; which is by listing them
in the same way we can list the integers: 0, 1, -1, 2, -2, 3, -3,
etc.

A ->list<- is a set X together with a bijection from either an initial
segment of the POSITIVE INTEGERS to X (or the naturals if you like
your lists to begin with 0), or from the entire set of positive
integers to X.

Every list corresponds to an indexed set, indexed by either N or an
initial segment of N. Not every indexed set is a list.

Form the OP:

"Let L_Omega be the list defined by the totality of all possible steps
of this procedure."

Your diagonalization procedure is certainly one possible step.

"If Arturo Magidin's step is a step then it is one of all possible
steps." is true. Just ask Virgil. So by definition, there is no
diagonalization steps remaining after they've all been performed.
There
can be no diagonalizing outside the set of all diagonalizations.
L_Omega can't be a list. It can't be diagonalized. It can't be
achieved by performing all the diagonalizations either, but that
doesn't mean it doesn't exist.

.

Relevant Pages

  • Re: Another stab at Cantor
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  • Re: Another stab at Cantor
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