Re: Cantor's wrong! (doubt it)

From: Russell Easterly (logiclab_at_comcast.net)
Date: 06/21/04 

Date: Sun, 20 Jun 2004 22:26:32 -0700

"Mike Terry" <news.dead.person.stones@darjeeling.plus.com> wrote in message
news:2jjk04F12j25hU1@uni-berlin.de...
> "Russell Easterly" <logiclab@comcast.net> wrote in message
> news:25KdnWyv0s2i5U_dRVn-gQ@comcast.com...

> Here is what I think you really meant to say:
>
> Definition: a natural number is "oney" if it's decimal representation
> consists entirely of a string of 1s.
>
> (e.g. 1, 1111, 111111 etc. are oney numbers)
>
> Definition: given the sequence {N_i}, we will construct a new *sequence*
> {x_i} as follows:
>
> x_i = (smallest oney number which is greater than every oney number
> appearing in the set {N_j: j<=i})
>
> (e.g. if N_i is the sequence 7, 2, 111, 5, 1111, ...
> then x_i is the sequence 1, 1, 1111, 1111, 11111, ...)
>
> This is the construction you were getting at, right?

Right.

> This is a properly defined sequence {x_j}, and we can show it is
> monotonically increasing. For each natural number j, you have derived a
new
> natural number x_j. Note you have not produced a number x_oo which is
> "right at the end of the list", as the list has no end...
>
> Now, depending on the original sequence {N_i}, the corresponding sequence
> {x_i} may converge to a final value (x), or may be unbounded in which case
> there is no final (x) to which it converges.

"Converges" is not the right word. Let's just say that the list, x, contains
a
natural number that is not in list N.

> So where do you want to go now? You seem to be suggesting that the
sequence
> {x_i} always converges to a number x, which is not in the original
sequence
> {N_i}. Well, sure, if {x_i} does converge to a number x, then x is
missing
> from the original list. BUT what if your {x_i} is unbounded?

My point is that these same objections can be applied to Cantor's diagonal
argument.

For example, let's use the following rule to create our diagonal number:
Let N be a list of real numbers and let x_i be ith digit of the diagonal
number.

If digit i of N_i = 1 then x_i = 2 else x_i = 1.
Let N be the following list of real numbers:

.000...
.1000...
.11000...
.111000.
.1111000...
...

x_i is "unbounded" and x never "converges" to a real number not in N.

My problem with the diagonal argument is that it requires an infinite
number of computations. There was a thread some time ago where
we proved that the Halting problem is solvable if we allow a
computer to perform an infinite number of calculations in finite time.
The diagonal argument requires such a computer.

Here is another example:
Consider a set of sets of natural numbers written in unary.

Unary numbering system:
One = 1
Two = 11
Three = 111
etc.

Sets of unary numbers:
N_1 = {1}
N_2 = {1,11}
N_3 = {1,11,111}
etc.

Notice that each of these sets contains a member whose length
is equal to the cardinality of the set. For example:

length of 1 = cardinality of N_1
length of 11 = cardinality of N_2
length of 111 = cardinality of N_3

Now consider a N that contains every natural number in unary.
Does this N contain a member whose length is equal to the cardinality of N?

Russell
- 2 many 2 count

Relevant Pages

  • Re: Cardinality vs Subsets
    ... set/sequence has a first and last member" or do you mean that the conditional in ... your criteria "Only if the set/sequence has a first and last ... In a standard sequence, every ... cardinality applies to all sorts of endless sets. ...
    (sci.math)
  • Re: An easy argumentation to show a severe limitation of set theory
    ... is the n-th element of the sequence S. ... members are, and you have not said whether N is a member or not. ... Then A is an infinite set, but N is not a member of A. ... cardinality). ...
    (sci.math)
  • Re: Uncountability of the Rationals?
    ... mentioned in the axioms, much less anything about omega or the alephs, ... The issue is whether cardinality CAN BE defined within an axiom system, ... but not inconsistent with cardinality ... How does TO define 'sequence' without reference to something like the ...
    (sci.math)
  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... fis the position in the old sequence from which you get the n'th ... For infinite sets of labels, ... > primarily about the cardinality of the continuum and continua, ... > cardinality, and that it could be explained that way. ...
    (sci.math)
  • Re: Mathematical concepts
    ... concept of counting to pre-schoolers. ... if you merely list all the ordinals ... applying ordinal names to each element in sequence, ... Regarding demonstrating that two sets have the same cardinality: ...
    (sci.math)