- 06 Oct '10 21:37

Are there more even numbers than natural numbers?*Originally posted by wolfgang59*

Of course!

I'm trying to point out the flaw in our opponents' argument in the "many balls" thread

Yet if I start with this sequence

{2} and {1,2}

{2,4} and {1,2,3,4}

...

{2,...,2n} and {1,2,...,n,...,2n}

...

n goes to infinity...

...

The difference in cardinality is always increasing in n!

Yet...the cardinality of the countably infinite sets is the same.

Now repeat after me "one-to-one correspondence"... - 06 Oct '10 22:31

yes but did you know that there are more evens than odds?*Originally posted by Palynka***Are there more even numbers than natural numbers?**

Yet if I start with this sequence

{2} and {1,2}

{2,4} and {1,2,3,4}

...

{2,...,2n} and {1,2,...,n,...,2n}

...

n goes to infinity...

...

The difference in cardinality is always increasing in n!

Yet...the cardinality of the countably infinite sets is the same.

Now repeat after me "one-to-one correspondence"...

for the odd nmber 1 there is 1+1=2

for the odd nmber 3 there is 3+1=4

for the odd nmber 3 there is 5+1=6

for the odd nmber Nthere is N+1

So for every odd there is an even ... but then we include ZERO

QED There are more evens than odds

- 07 Oct '10 04:28

inf - inf is not zero. It's not even defined. Therefore your calculation is flawed.*Originally posted by wolfgang59***1+2+3+4+ ... equals?**(1+1+1+1+...)

ZERO

Obviously 1+2+3+4+ ... = (2-1)+(3-1)+(4-1)+(5-1)+...

=2+3+4+5+... [b]-

= inf - inf

= 0[/b]

If we define inf - inf, inf / inf, and such, as a number then we get funny results, as the one above. - 08 Oct '10 10:21

So which standard operation breaks down when you reach an infinite sum? Could you get that result by extrapolating from the union of finite sequences of sums?*Originally posted by Thomaster***Did you know that 0 = 1?**

0 = 0 + 0 + 0 + ...

= (1-1) + (1-1) + (1-1) + ...

= 1 + (-1+1) + (-1+1) + (-1+1) + ....

= 1

QED - 08 Oct '10 10:37 / 1 editIf I remember correctly, it's all to do with whether the series is absolutely convergent or not.

SUM(x_i) is absolutely convergent <=> SUM(|x_i|) is convergent.

Absolutely convergent series behave when you rearrange them. Series that are just convergent (e.g. 1 - 1/2 + 1/3 - 1/4 + ...) can converge to different values (or not at all) depending on the order.

For example:

1 - 1/2 + 1/3 - 1/4 + 1/5 +... = ln 2

1 + 1/3 + 1/5 + ..... - 1/2 - 1/4 - 1/6 - ... does not converge

1 - 1 + 1 - 1 + ... is very obviously not absolutely convergent!