Re: Cantor Confusion
- From: WM <mueckenh@xxxxxxxxxxxxxxxxx>
- Date: 21 May 2007 07:11:03 -0700
On 21 Mai, 13:47, stevendaryl3...@xxxxxxxxx (Daryl McCullough) wrote:
WM says...
On 20 Mai, 17:57, stevendaryl3...@xxxxxxxxx (Daryl McCullough) wrote:
WM says...
> It is. Set theory is simply biased. Consider the list
> 0.666...
> 0.3666...
> 0.33666...
> 0.333666...
> ...
> If the diagonal number is defined by "replace 6 by 3", then we have
> two answers none of which can be preferred by logic, but the second of
> which is suppressed by convention.
The diagonal number is 0.333... which is not on the list.
The diagonal number cannot have more 3's than every list number has
3's.
Of course it can.
Consider the following infinite matrix M:
xooo...
xxooo...
xxxooo...
....
You claim the (unchanged) diagonal can have more x's than any line?
Than is wrong because the diagonal consists of line elements.
With your rule for the diagonal: replace 3 by 6, then
if we start with the following list:
0.6000
0.06000
0.006000
0.0006000
0.00006000
...
then the diagonal will be the same: 0.333...
There is one "3" for each number in the list.
What do you think why I chose the special list? In that list the
diagonal number cannot have more 3's than at least one line of the
list (well, one more).
If the diagonal number is complete in the sense that at every
position indexed by a natural number there is a 3, then there must be
a complete sequence of 3's among the list entries too.
No, that's false. It's *provably* false.
It is also provably right. See matrix M above.
For your particular
example, what is true is this (letting r_j be the jth number
in the list)
1. For each natural number n > 0, there exists an index j
such that r_j agrees with the diagonal number in the first
n decimal places.
2. For each index j, there exists a natural number n > 0
such that r_j does *not* agree with the diagonal number in
the first n decimal places.
Statement 1. says that the diagonal number can be approximated
arbitrarily well. Statement 2. says that the diagonal number does
not occur on the list. Both are true.
But they are contradictory, because more than "each natural" number is
not available. Forall n in the first n decimal places means for *all*
natural indexes.
These two statements have the logical forms:
1. forall n, exists j, Phi(r_j, d, n)
2. forall j, exists n, ~Phi(r_j, d, n)
where Phi(r_j, d, n) is the statement "r_j agrees with the diagonal
number d in the first n decimal places".
Otherwise the list is not complete, i.e., there is always a 6 at
some natural index.
The definition of the diagonal number d is the following: (letting
r[n] mean the nth decimal place of the real r)
d[n] = 9 - r_n[n]
The definition of the numbers r_j is this:
r_j[n] = 6, if j=n
= 3, otherwise
It follows that
r_n[n] = 6
Putting those together, we have
d[n] = 9 - 6 = 3
So every digit of the diagonal number is 3.
If there are infinitely many 3's in the diagonal, then there are
infinitely (- 1) many 3's in an r_n.
If the list is not complete however, the diagonal number cannot be
complete either.
The list is "complete" in the sense that for every natural
number j, there is a corresponding real number r_j. Similarly,
the diagonal number is complete in the sense that for every
natural number j, there is a corresponding digit d[j].
And the saying that at *every* position there is a
digit b_n =/= a_n,n is nonsense.
It's provably true. (I'm using d, rather than b, and I'm
using r_n instead a_n).
d[n] = 3
r_n[n] = 6
Clearly d[n] is never equal to r_n[n].
That is only one side of the medal which does not take into account
that hat each real number r_n is not better determined than by |r_n -
r_n(k)| < epsilon(k), or to express it opposite: An irrational real
number r_n cannot be determined better than by a sequence r_n(k) with
|r_n - r_n(k)| > anti-epsilon(k).
In other words: An irrational real number *is not* defined better than
up to an anti-epsilon > 0 because for every approximation r_n(k) we
can find such anti-epsilon > 0 which is less than the difference |r_n
- r_n(k)|.
For the entries E(n) of the list we find
lim[n-->oo] (E(n) - 0.333...) = 0.
It is the same case as lim[n-->oo] (1 - 0.999...9 with n 9's) = 0.
That's true. In this particular case, the limit of the sequence
is equal to the diagonal of the sequence. So what?
Why only in this particular case? Is lim[i --> oo] (b_i - a_i) * 10^-i
= 0 correct only in one special case?
I say in this case because not every sequence has a limit. For example,
the sequence
0.600...
0.330...
0.6660...
0.33300...
0.666600...
0.3333000...
has no limit. But it still has a diagonal, namely
0.363636...
But if this diagonal is to be considered a real number, then we need
the factors 10^-i. The same is true for
lim[i --> oo] (b_i - a_i) * 10^-i
If every initial segment of the diagonal number is represented by the
initial segment of an entry of the list, then the full diagonal number
is represented by an entry of the list.
That's false.
No, that's correct.
It's provably false. The diagonal number in your case satisfies
d[n] = 3
Every element has r_n[n] = 6. Clearly they are not the same, so
d is not equal to r_n.
It is the only valid interpretation of the notion infinity.
I don't know what that's supposed to mean, but you seem to be
confusing three different concepts:
1. The number d appears on the list r_j. For this to be true,
it must be the case that Exists j, Forall n, d[n] = r_j[n].
That is *false*. Provably false.
No. It may be provably false for the diagonal sequence, but not for
the diagonal number.
2. The number d is the limit of the numbers r_j. For that to be
true, it must be the case that Forall n > 0, Exists j, Forall m < n,
d[m] = r_j[m]. That's true in your case.
And that is *all* we can consider in a list f n-ary expansions. Any
infinite n-ary expansion of a real number (not ending by zeros) *is
not* defined better than up to an anti-epsilon > 0 because for every
approximation r_n(k) we can find an anti-epsilon > 0 which is less
than the difference |r_n - r_n(k)|.
To say that the number r appears on the list r_0, r_1, ...
is to say that there is some natural number j such that r = r_j. If
we let D_j = |r - r_j|, then the criterion for r appearing on the list
is that
exists j such that D_j = 0
This criterion is false.
If D_j is not zero, then that means that the diagonal number
is not equal to r_j.
In case lim [i --> oo] (10 - 9) * 10^-i = 0 we see it clearly.
What do limits have to do with anything? There are two different
questions:
1. Does the diagonal number d appear on the list? The answer is no.
This answer is given at most for the diagonal sequence, not for the
diagonal number.
2. Is the diagonal number the limit of the numbers on the list? The
answer for your case is yes.
And that is *all* we can consider in a list f n-ary expansions. Any
infinite n-ary expansion of a real number (not ending by zeros) *is
not* defined better than up to an anti-epsilon > 0 because for every
approximation r_n(k) we can find an anti-epsilon > 0 which is less
than the difference |r_n - r_n(k)|.
Forall n e N we have 0 < 1 - 0.999...9 (with n 9's).
Yes, that's true. 1 does not appear on the list
0.9
0.99
0.999
0.9999
0.99999
etc.
and 0.33333 does not appear on the list
0.6666...
0.36666
0.336666
0.3336666
etc.
So r does not appear on the list.
So the problem of replacing 0 by 9 is not existing.
What are you talking about? Of course the number 0.9999...
(all 9s) exists. And it is equal to 1. But it is not on the
list
0.9
0.99
0.999
0.9999
0.99999
etc.
But we have in the limit j --> oo: D_j = 0.
Yes. That's true, but irrelevant. For the diagonal to be
on the list, it must be that D_j = 0 for some finite j.
That is only required for a finite list.
It's not good enough that limit of D_j = 0.
Consider Dave L. Renfro's example:
0.49999999...
0.49111111...
0.44911111...
0.44491111...
0.44449111...
.. . . . . . .
Replacing 4 by 5 and 9 by 0 gives the diagonal number
0.5000...0999... =/= 0.5 for any finite j. Only in the limit j --> oo
the diagonal number is 0.5000... = 0.499..., the first entry. Here it
is good enough to consider the limit to make the proof fail?
Regards, WM
.
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