Infinity and Paradoxes

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Infinity and Paradoxes

Postby undecim » Mon Feb 21, 2011 4:25 pm UTC

I'm looking for interesting paradoxes involving infinity.

One example I thought of last night:

[math]\sum_{k=1}^{\infty}k - \sum_{k=1}^{\infty}2k-1 = 2\sum_{k=1}^{\infty}k[/math]
[math]\left(1 + 2 + 3 + 4 + \cdots\right) - \left(1 + 3 + 5 + 7 + \cdots\right) = 2 + 4 + 6 + 8 + \cdots[/math]

In other words, if you subtract all the odd terms from the series, that's the same as doubling all the terms.

As I understand it, this is roughly equivalent to the Banach-Tarski paradox

EDIT: sorry, I failed at basic arithmetic... This doesn't work with the harmonic series
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Re: Infinity and Paradoxes

Postby LLCoolDave » Mon Feb 21, 2011 4:42 pm UTC

The difference is anything but well defined, it depends massivly on how you put your brackets. Conversly, I could argue that each number in the second series is at least as big as the corresponding one in the first sum, so I get

[math](1-1)+(2-3)+(3-5)+(4-7)+...= \sum_{k=0}^\infty -k = - \infty[/math]

The issue is that the difference of these two series is not well defined, depending on just how I construct the differences of partial sums I can get very different results in the limit, and there is most certainly no "one correct" way to do it.

This is in no shape or farm related to the Banach Tarski Paradoxon at all, it's just an ill defined term (and therefore a nonsensical conclusion).

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Re: Infinity and Paradoxes

Postby z4lis » Mon Feb 21, 2011 5:51 pm UTC

Yeah, I wouldn't say that this is roughly equivalent to the Banach-Tarski paradox... however, I would say that they have in common the idea of paradoxical decompositions of "infinity". At any rate, if you liked what you came up with, you would probably really be blown away by what Riemann had to say about bad series:

It basically says that if a series converges, but that's because of cancellation (that is, the sum of the absolute values of the terms goes to infinity), then if you give me any real number, I can rearrange the terms of the original series to add up to that number. I can also rearrange them to go to infinity or negative infinity. It's a pretty strong indication about how commutativity breaks down under taking limits. :P

If you want more stuff on infinity and its paradoxes, there are plenty of interesting philosophical and mathematical questions floating around on it, since we've been collectively wrestling with the concept since its inception. If you haven't already, you might also want to read some set theory and learn about cardinality, where you'll see some statements about how there are different "sizes" of infinity. An "infinite" number of different sizes, of course. :P
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Re: Infinity and Paradoxes

Postby Eastwinn » Mon Feb 21, 2011 6:45 pm UTC

What you've shown is manipulation of symbols without meaning. A divergent sum doesn't actually represent a value, so how can you subtract them? Just another example of how intuition seems to fail us when dealing with infinities. - surreal sketches and characters.

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Re: Infinity and Paradoxes

Postby doogly » Mon Feb 21, 2011 8:21 pm UTC

Here's your problem, you forgot that
[math]\sum_n n = -\frac{1}{12}[/math]
From there it should be easy to see where to go.
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Re: Infinity and Paradoxes

Postby ImTestingSleeping » Wed Feb 23, 2011 1:03 pm UTC

You can't rearrange a series which isn't absolutely convergent.

If you are looking for something strange, then I would offer the fact that for any conditionally convergent series [imath]\sum a_n[/imath], for any number x there exists a rearrangement of [imath]\sum a_n[/imath] such that [imath]\sum a_n[/imath]=x.

[Edit] Whoops, didn't notice z4lis already linked Riemann's theorem in an above post. Neat stuff.

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