Re: Infinity does exist?

From: |-|erc (gotcha_at_beauty.com)
Date: 06/30/04 

Date: Wed, 30 Jun 2004 01:46:45 GMT

"Hasan Demir" <has_andemir@yahoo.com> wrote in
> You add 1 to a infinite number bla bla bla... if we should add 1, it
> is not an infinite number, it has an and! X will be one more big than
> (X-1), nothing but this. We call amounts that we can not calculate or
> think about (whatever), we call it infinite.
> ------------------------
> Thanks for reading!

Right! Unlimited and infinite should not be seperated, much the same meaning,
if you objectify infinity like mathematicians then of course you will think you know of BIGGER infinities.

Imagine all the rational numbers, a/b, where a and b are integers.
Then plot them on a 2D chart

          Numerator 1 2 3 4 5 6 ...
Denominator
1 1/1 2/1 3/1 4/1 5/1
2 1/2 2/2 3/2 4/2 5/2
3 1/3 2/3 3/3 4/3 5/3
...

This chart covers every rational number.

Its looks like there's more rational numbers than integers, since they only
take up a line, and there's a whole plain of rationals. Are there MORE rational numbers
than integers?

No, because you can COUNT them *just like integers*.

1 <=> 1/2
2 <=> 2/1
3 <=> 1/2
4 <=> 3/1
5 <=> 2/2

simply by drawing along increasing diagonals, like so.

1 2 4 7
3 5 8
6 9
10

By this chart the 7th rational number is in position (x,y) = (4,1) => 4/1 = 4.

So there are really just as many rationals as integers!

What about irrationals? SQRT(2), e, pi, cos(.2), ...

This is Cantors proof :

Line up the list of numbers like we did with the rationals.

1 <=> 1/2
2 <=> 2/1
3 <=> 1/2
4 <=> 3/1
5 <=> 2/2

Only this time, the numbers are non terminating, like pi = 3.14159265....
never stops with new digits sequences, so we have to draw all the digits.

1 <=> 0.346435436..
2 <=> 0.674376474..
3 <=> 0.1112123333..
..

Is this COUNTABLE list complete?

No, because we can form a new number. Look at the 1st digit of the 1st number,

0.3

and the second digit of the second number, 0.67

This makes the diagonal number 0.371...

Then add 1 to each digit and you get 0.482...

0.482... CANNOT be anywhere on the countable list, because every number on
the list is different in at least one digit!

Therefore, the only conclusion they could think of was there must be HIGHER INFINITY
than countable infinity.

Unfortunately until a better theory comes along, its put a halt on the usefullness of computers
since mathematicians are INTENT they can do heaps of miraculous theories computers
can't handle.... since every thing a computer can do can itself be counted, so they seem stuck
with integers.

This is where they are going wrong :

<
<1>
>

the sequence <2> is not in the list
:: the list is missing a finite sequence of digits

<
<12>
<34>
>

 the sequence <25> is not in the list
:: the list is missing a finite sequence of digits

<
<123>
<456>
<789>
>

the sequence <260> is not in the list (ignore recurring 9s)
:: the list is missing a finite sequence of digits

Case n
<
<xx..3>
<xx..6>
..
<xx..9>
>

 the sequence <yy..0> (the mod_diag) is not in the list
:: the list is missing a finite sequence of digits

IMPLIES

Case n+1
<
<xx..3a>
<xx..6b>
..
<xx..9c>
<xx..0d>
>

 the sequence <yy..0e> (the mod_diag) is not in the list
:: the list is missing a finite sequence of digits

THERFORE
<
<x>
<xx>
...
>
This list is missing a finite sequence of digits.

This is what they concluded from Cantors proof. When the correct induction should be

<
<xx..3>
<xx..6>
..
<xx..9> row n
>

For All finite sets of numbers, there are missing numbers.

This is evident when you consider that the computable list of numbers will
generate every sequence of digits in time, being infinite means its unlimited.

Herc

Relevant Pages

  • Re: Infinity does exist?
    ... > think about, we call it infinite. ... never stops with new digits sequences, so we have to draw all the digits. ... :: the list is missing a finite sequence of digits ...
    (sci.math)
  • Re: Infinity does exist?
    ... > think about, we call it infinite. ... never stops with new digits sequences, so we have to draw all the digits. ... :: the list is missing a finite sequence of digits ...
    (sci.logic)
  • Re: abundance of irrationals!)
    ... > There we have all numbers with a finite sequence of digits 1. ... Every number in the list has the same (countably infinite) ... the list is incomplete. ...
    (sci.math)
  • Re: limitation to induction on finite bounds
    ... :: the list is missing a finite sequence of digits ...
    (sci.math)
  • Re: limitation to induction on finite bounds
    ... :: the list is missing a finite sequence of digits ...
    (sci.logic)