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I am reading an e-book called To Infinity and Beyond by Dr. Kent A Bessey. In the book the author makes the claim that Georg Cantor made a discovery "where half of a pie is as large as the whole".

In talking about it, he seems to claim that because half a pie can be broken into an infinite amount of pieces, and likewise a whole pie can be broken into an infinite amount of pieces they are infact the same size.

By the same concept, he states that if you took all of the pieces of the edge of a box you could create as many more boxes of whatever size you wanted using those pieces.

This seems undeniably false to me. I cannot help but draw a parallel between limits -> infinity. Where those limits may equal 2 or some other finite value. In my view, even if you were to break half a pie into an infinite amount of pieces the pieces could never add up to more than half a pie.

Am I misunderstanding? Can someone explain this concept better?

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This sounds a bit like a garbled version of en.wikipedia.org/wiki/Banach-Tarski_paradox. Cantor certainly recognised that an infinite set can be in bijection with one of its proper subsets (though I doubt he was the first); nothing to do with the size of pies, though. –  Asal Beag Dubh Feb 11 at 15:58
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The set of even natural numbers can be put in one to one correspondence with the set of natural numbers. This was observed by, among others, Galileo. There are similar results for other infinite sets. For advice about pies, perhaps one could go to the site Seasoned Advice. –  André Nicolas Feb 11 at 16:01

5 Answers 5

up vote 24 down vote accepted

There are (at least) two kinds of "size" in mathematics. One is cardinality. In set theory, the cardinality of a set is the number of elements it contains, and two sets have the same cardinality if there is a one-to-one mapping between them. This is a coarse kind of "size", in that many different sets share the same cardinality: the interval $[0,1]$ is the same size as the entire real line, which in turn is the same size as all of $\mathbb{R}^3$. In this sense, which is likely to be Cantor's meaning, half of a pie is certainly the same size as a whole pie. Another type of "size" is measure, which assigns real numbers to (some) sets in such a way as to generalize the usual lengths of line segments, areas of polygons, volumes of cubes and spheres, etc. This is much more precise: if you cut a disc of area $1$ into measurable pieces and then reassemble those pieces however you like, the result will still have area $1$. However, if you allow any type of pieces (not just measurable ones), then the Banach-Tarski paradox can happen: a sphere of volume $1$ can be cut into a finite number of pieces and reassembled into a sphere of volume $2$. (This can't happen in the plane, though, so your planar pie is safe.)

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The pies are a bad analogy, since they aren't infinite. However, maybe the following analogy would be useful and a different perspective from other answers:

Suppose a genie gives you a magic cup of beer that's always full, no matter how much you pour out or drink of it. Now, let's suppose that you get another, exactly identical cup - can you argue that the total amount of beer inside of them both is different than the amount of beer in the first one?

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I like this. Using a conceptual analogy to explain an idea that is purely conceptual. –  Cruncher Feb 11 at 21:08
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I guess the only real thing to consider here is that now I can drink beer twice as fast. –  Cruncher Feb 12 at 15:56
    
It doesn't work for real pie, only thought pie. Bummer. –  Daniel Cook Feb 12 at 19:10

If you want your mind blown some more, try this.

As for what is going on here, an infinite set is always the same size as half of itself because we can define a bijection between the two. The problem you seem to be running into is the author trying to give examples of how this might work using things that don't really conjure an image of the infinite. I would love to have an infinite size pie that I could cut some from and always have more. That being said, it is not exactly a good image for dealing with the infinite. The problem is, infinity is a concept and can never be represented exactly by something we can assign a number to. Better (as in larger sized numbers) examples might be grains of sand, stars in the sky or atoms in the universe. Though these examples are a little better, they still don't represent the fact that infinite sets are the same size as half of themselves.

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So, is that how the TARDIS works? –  JDB Feb 11 at 17:56
    
@JDB Could be but the Doctor doesn't seem to be willing to let us know if that is true. –  John Habert Feb 11 at 17:59
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I don't think the example of the sets with big numbers helps at all actually. If I take half the sand away from the beach, I observe the exact same kind of change as if I eat half of a pie. Half is gone. This only works with infinity, and there is no in between. –  Cruncher Feb 11 at 21:04
    
@Cruncher That is why I say they don't really represent the concept of the infinite. But it is a bit easier to imagine a pile of sand split into two piles as big as the original (since there is lots of sand) compared to a pie. –  John Habert Feb 11 at 22:32
    
I think an infinite pie is a great image for dealing with the infinite. It's just misleading as you tend to accidentally assume you're dealing with a real pie. Indeed, much of the description in that link goes quite over my head as I have little Math experience beyond Pre-Calculus. I did find the simplified description which follows this comic helpful. –  Daniel Cook Feb 12 at 19:14

The mass of the half pie is always smaller than the mass of the whole pie. Your point of view is related with the mass (in mathematics that can be calculated by integrals), but cardinals are something else. You may refer for that to many articles or even documentaries on the internet.

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In my personal opinion, I would explain as follows.

  1. half of a pie < whole pie.

  2. half pie ---process---> infinity pieces.

    For the same reason, whole pie ---process---> infinity pieces.

  3. But process need ENERGE.

  4. And I SUPPOSE that half pie need lots of energe to be infinity pieces.

    For the same reason, whole pie need lots of energe to be infinity pieces.

    I think that lots of energe = ∞.

  5. So, half pie + lots of energe = half pie + ∞ -> infinity pieces

    whole pie + lots of energe = whole pie + ∞ -> infinity pieces

    by my understanding:

    half pie + ∞ = whole pie + ∞

    ∞ = ∞

  6. If someone can slice the half pie into infinity pieces, I think the guy could easily restore the pieces to a pie as big as the whole pie.

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This explanation might be good if I could understand it. You start to lose me at ", if PieA === PieB(mass)". By step 4 you've completely lost me. –  Daniel Cook Feb 12 at 19:17
    
My fault. I edited it. I tried to explain my opinion more clear. –  c-s.c Feb 12 at 21:12

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