Re: Cantor's "diagonal argument". My Objection.

LudovicoVan schrieb:
On 6 Dec, 17:46, Jan Burse <janbu...@xxxxxxxxxxx> wrote:

There are
many many(*) orders on the natural numbers.

Here I give you an order of type omega+omega of
the natural numbers:

1 3 5 ... 2 4 6 ...

This is very interesting. A couple of questions if you don't mind:

1) I take it to be something like: 1,3,5,...|2,4,6,... How would you
define this sequence formally? (I am puzzled by the non-ending in
between);

In this case I can do it formally based
on the natural numbers. Let J denote
the ordering on NxN. We can define it as
follows:

J(n,m) :<=> if odd(n) & even(m) then true
else if even(n) & odd(m) then false
else n<m

2) What is the cardinality of this sequence (should I say, of the set
of the elements of the sequence)? I guess it is still Aleph_0, but I
think I'd need the formal definition of the sequence to see exactly
how.

The cardinality of the domain of J is the same as
the one of the natural numbers, as the domain
are the natural numbers.

(*) Actually an order on the natural numbers
can be coded in a real number. Guess why?

This is just as interesting: could you please elaborate a little bit?
I can't guess how that is.

We need to show that *any* J can be coded
by a real r.

First of all we code the pairs as single natural
numbers. Thus J(n,m) becomes H(<n,m>), where <.,.>
is an injective mapping from NxN to N. There
are many such mappings available. The simplest
mapping is based on traversing the grid:

+------------
|0 1 3
|2 4
|5
|

<n,m> = (n+m)(n+m+1)/2 + n


Then we see that H is in P(N), i.e. the power
set of N. Now use any injective mapping from P(N)
to the reals R. This is a little bit tricky.
We basically use only the interval [0,1] and
the binary development.

So there is a binary digit 1 after the point
at position n, if H(n) holds. And there is
a binary digit 0 after the point at position
n, if H(n) does not hold. So for example
the finit H={3,5} gives the following real:

0.00101

Also infinite H's can be coded. But we are
not yet finished. We face the following
problem. For examples the following binary
developments denote the same real:

0.00100000...
0.00011111...

So we will actually map to the interval
[0,2], and when ever we have an ending
1111 we add 1. So the above two would be
coded as:

0.00100000...
1.00100000...

Now we are done, we have an injective
mapping from any relation J to [0,2].
The orderings are a subset of all the
relations, and are thus also mapped.

Bye
.

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