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This is a simple question, and maybe stupid:

Is this true:

$\infty < 1000\cdot\infty$ ?

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10  
In various contexts the RHS isn't well-defined. In some contexts where it is, this is true, and in some contexts it's false. –  Qiaochu Yuan Aug 5 '11 at 21:14
4  
It is not a simple question, nor is it stupid. –  André Nicolas Aug 5 '11 at 21:20
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It's hardly philosophy, though. –  lhf Aug 5 '11 at 21:32
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@user11775: In my opinion, the statement "The value of a life is $\infty$" doesn't mean anything (nor would "The value of a life is $17$" for that matter), and even if it did, there's no way it could be "determined" to be true or false. The fact that humans have decided to make certain definitions does not imply anything about reality. –  Zev Chonoles Aug 5 '11 at 22:13
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@user11775: I am very much against the abuse of mathematics as a rhetorical and usually demagogic tool by people who don't fully understand the concepts they refer to. Please don't equate the value of life to infinity. –  Asaf Karagila Aug 5 '11 at 23:04

1 Answer 1

up vote 10 down vote accepted

It's definitely not simple or stupid.

You hear a lot of stuff floating round about $\infty$ - some true, some false, some purely conjectural, some more philosophical than mathematical, but it's almost all subject to the same flaw: it doesn't really tell you what infinity is. And that's because that's not an easy question to answer. When someone says "is $\infty = \infty + 1$?", the only correct and reasonable answer is "what do you mean by $\infty$?". But of course, the nature of $\infty$ is such that anyone who knows what they mean by $\infty$ knows the answer to these questions. So here are a few examples of what $\infty$ can mean - and it will all vary by context.

  1. $\infty$ might just be a symbol. Often people will modify the complex plane by adding on an extra point "at $\infty$", i.e. infinitely far away from $0$ in any direction. This sounds like an odd thing to do (because the $\infty$ you get to by going off in one direction is the same as the one you get to in any other direction!), but it's surprisingly useful. So useful, in fact, that it's got its own name: compactification. It gives you a very geometric way of interpreting things called Möbius maps, which are useful all over the place, from number theory to geometry, but which without this interpretation would just look a bit horrible. They're even useful for seeing what's going on in some Escher paintings! This one has a simple answer: things like $\infty+1$, $1000\times \infty$, $2^\infty$ etc. don't make sense.

  2. $\infty$ might be an attempt to replicate an "arbitrarily large number". For example, you might stick an extra point called $+\infty$ on the right of the real line, and one called $-\infty$ on the left. In practice, we don't do this, we invoke something called limits, but the answer, as above, is that it doesn't make sense.

  3. It might count the size of a set. If you take a set, e.g. $\{a,b,c\}$, you know it has three elements by matching up each element with a number (1, 2, 3, ...) until you have to stop. This gives us a nice way of determining whether a set is infinite or not: count the elements until you have to stop, and if you never stop, it's infinite! More sensibly, it tells us when two sets are the same size - if you can pair off their elements one-to-one, then they're the same size. We normally show that $A$ and $B$ are the same size by picking a function $f:A\to B$ which matches them up one-to-one, i.e. has an inverse function $f^{-1}:B\to A$. So $\{0,1,2,\dots\}$ and $\{1,2,3,\dots\}$ are the same size ($f(x) = x+1$). Importantly, $\{1,2,3,\dots\}$ and $\{1000,2000,3000,\dots\}$ are the same size ($f(x) = 1000x$). But obviously there are intuitively 1000 times as many elements in the first set as in the second. So here $1000\times\infty = \infty$, and your answer is no.

  4. It might be an ordinal. It doesn't seem to make sense on the face of it to "count past infinity" - certainly not where sets are involved. But there is a context in which it can be made rigorous. Consider the sequence of points on the real line 0.1, 0.11, 0.111, 0.1111, ... - this thing never ends, and clearly it has infinitely many points, with a limit point at 0.111... = 1/9. What happens if I put an extra point at 0.2? Then, travelling from left to right, we have to get past this whole infinite limit and we still have an extra point. Somehow we have to treat isolated points and limit points differently so that we have a way of getting "past" an infinity, but it can be checked that this does all work rigorously, and gives a sense in which $\infty$ and $\infty + 1$ might be different. (Problem: we had to choose a direction to travel in, and as a result of this, addition now doesn't commute, so $1 + \infty = \infty \neq \infty + 1$!) In this context, the answer to your question is yes.

I strongly recommend you go and search it on Wikipedia, or maybe read Brian Clegg's book "Infinity" - these may give you some ideas of what's going on in a mathematician's mind when you mention $\infty$.

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5  
Billy: I am not a fan of the treatment you gave to ordinals. The explanation is unclear about what an ordinal is. Also ordinal multiplication is not commutative, and for example $1000\cdot\omega=\omega<\omega\cdot 1000$. –  Asaf Karagila Aug 5 '11 at 21:55
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Sure. But my post was not a treatment of ordinals. (Deliberately so, which is why you won't find a clear definition of what an ordinal is in my post!) It was a quick explanation of how it's possible to count "past" infinity; introducing too many subtleties and too much machinery would have made my post unreadable to anyone other than a final-year undergraduate or up. Teaching is not just exposing information, it's all about making sure the other person learns, so I subjected myself to the constraints that the ordinal stuff should be intuitive and no more than a paragraph. :) –  Billy Aug 5 '11 at 23:11
    
Billy: I am having a hard time in finding intuition with regards to ordinals there. My favorite analogy to develop intuition is queues. Empty queue, finite queue, $\omega$ queue (an infinite line, but every one has only finitely many people before him), and then we add a poor guy who has to wait until all the infinitely many people are done and only then his turn will arrive - i.e. $\omega+1$ queue. Addition and multiplication becomes clearer that way as well, as queuing several copies, or "pushing back" one queue by another... and so on. –  Asaf Karagila Aug 5 '11 at 23:15
    
As I said, I wasn't teaching ordinals. It was related, but it certainly wasn't a treatment of ordinals. I dropped the word in as something to google, and gave an illustration of approximately how they behave. Anyway, I've not heard your analogy before - I like my example of points rather than people perfectly well, but yours is probably friendlier. If you want to go ahead and teach ordinals properly, post your own answer! –  Billy Aug 6 '11 at 1:48
    
@Asaf Karagila: I am amused that I upvoted both Billy's answer and your comment. I think that is consistent. I think both will contribute to understanding of OP and others. –  Ross Millikan Aug 6 '11 at 4:16

protected by Qiaochu Yuan Aug 5 '11 at 21:59

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