When inifinity isn’t infinite

By Marcus Wilson 05/02/2013 7

I’ve been having some discussion with a collaborator in Sydney regarding a numerical model that we are developing. It concerns the response of the brain to pulses of magnetic field, but for the purposes of this blog entry, that is immaterial. One thing that we’ve been grappling with is ‘dealing with infinity’. Basically, in physical processes, what is happening now depends on a sum of everything that’s happened previously. Processes are causal. To understand the present, we look at the past (not the future). How far back in the past do we have to look? That depends on the problem, but, technically speaking, we have to look infinitely far back in the past. There is not cut-off time. Sure, there comes a point where what happened X seconds (or hours, years, millennia ago) isn’t going to make much of a difference, but where we take this cut-off is usually arbitrary. So often we physicists simply say ‘start from a time of minus infinity’, which means consider all things in the past, no matter how far back they happened.

What we get are integrals that run from minus infinity to now in time. And that, in the case of the model that I’ve been looking at, causes some conceptual problems. We end up with an infinite result. But that doesn’t actually turn out to be a physical problem. That’s because what we actually want to know involves integrating this result over another infinite time range, which generates an infinite sum of infinities. But here’s the neat bit. Some of these are positive infinities and some are negative infinities, and they cancel and divide out to something sensible and finite. It sounds odd, but mathematically it is all quite reasonable.

There are a few examples of this kind of thing in physics. Perhaps the most significant is renormalization in particle physics. Here, when one calculates using quantum electrodynamics how electrons and photons interact, the calculations are full of infinities. But the infinities, if considered carefully can be shown to cancel out. The positive infinities cancel with negative infinities. It was a major conceptual leap forward to accept that this sort of thing was reasonable.

I first came across the idea with my PhD – looking at how electrons interact with each other in solids. Here, there is a seminal paper by Gell-Mann and Bruekner (1957) which shows that while perturbation theory gives an infinite sum of infinite integrals, the infinities can be dealt with and a finite result remains. After a bit of on-line hunting I tracked it down here. It’s rather neat (if you’re a physicist).

The moral of all this for the non-mathematician is to be careful with the idea of infinity. Don’t treat infinity as ‘the biggest possible number’. There is no such thing. If you think you’ve found it, let me know and I’ll prove you wrong by adding one to it.  It’s simply a short-hand for ‘and keep going…’


7 Responses to “When inifinity isn’t infinite”

  • “Don’t treat infinity as ‘the biggest possible number’. There is no such thing. If you think you’ve found it, let me know and I’ll prove you wrong by adding one to it.”

    That won’t work with transfinite numbers. Adding 1 to aleph 0 just gives you aleph 0 back again. It is possible to construct bigger numbers, it just takes a little more than simply adding 1. See http://en.wikipedia.org/wiki/Transfinite and http://en.wikipedia.org/wiki/Beth_number

    • You might have to help me on this one. My maths isn’t up to it. Pure maths is one of the reasons why I’m a physicist not a mathematician. Dealing with an infinite sum over infinite integrals that has a finite result is tough enough!

  • OK well I’ll try and write something that I hope will help and apologise in advance if at any point you feel like I’m talking down to you. :)

    Firstly I absolutely concur that in calculus and sequences when you have a sum or integral to infinity it isn’t a number it’s just shorthand to take the unbounded limit.

    On the other hand say we consider numbers as corresponding to the sizes of sets. We would then define the size of a set {x} as 1, {x, y} as 2, {x,y,z} as 3 and so on. Two numbers will be called equal if we can find a bijective map between sets that represent them (I’ll get back to why this is somewhat important).

    Now we have done this it’s not entirely unreasonable to ask ‘what is the number that represents the size of the set of natural numbers?’. I *think* it’s *defined* as aleph 0 (usually written as the hebrew letter aleph with a subscripted zero but I can’t really do that here). Since it’s a definition I expect there are some interesting arguments about it but that’s something I don’t know about. Aleph 0 is also sometimes called ‘countably infinite’.

    The next step then is to ask what other sets are countably infinite? This brings us back to the bijective map. In the case of countably infinite numbers being able to list each one in an infinitely long list is equivalent to having a bijective map with the natural numbers. So for example the integers are also countably infinite as I can list them all by 0, -1, 1, -2, 2, -3, 3, … and every integer will be in that list somewhere. The rational numbers are also countably infinite but that’s easier to demonstrate with a whiteboard sorry, you demonstrat it by creating a table and then going diagonally across the table in slices to get all rational numbers. 😛

    So to your proposal to just add one to these transfinite numbers the reason it won’t work is that it is equivalent to taking a set like the natural numbers and adding an element, say x, to it. But I can still make a countably infinite set out of it by listing the elements as x, 1, 2, 3, … I can in fact add a countably infinite number of elements to a countably infinite set and still obtain a countably infinite set out of it (Out of x_1, x_2, x_3, … and y_1, y_2, y_3, … make x_1, y_1, x_2, y_2, x_3, y_3, …).

    A subsequent question that can then be asked is if there are in fact different sizes of infinite numbers. The answer is yes! The real numbers are generally considered to be the next size of infinite called uncountably infinite or the continuum (I’ll come back to aleph and beth numbers in a moment). The proof that the real numbers are a different size of infinite is actually pretty cute. Suppose you have a complete list as described above of all real numbers. Then build a number by taking the ith digit after the decimal point of the ith number in the list and just use a different number in the number you are constructing. You will then have constructed a number that was not in the original list, as it differs with the ith number at the ith decimal place, but is still a real number thus contradicting that the list was complete to start off with. Thus the size of the set of real numbers is bigger than the size of the set of natural numbers.

    As for a general construction you can always build a bigger set, even in this infinite case (and assuming some basic axioms hold) by taking the power set (set of subsets) of a set. This construction gives the beth numbers (again hebrew letter subscripted). beth 0 is the natural numbers, beth 1 is the size of the set of the power set of the natural numbers (and is the size of the set real numbers), beth 1 is the size of the power set of the power set of the natural numbers (this is equivalent to the size of the set of all functions from real numbers to real numbers), and so on. There is a beth number for every natural number, so not only is there more than one size of infinity, there is a countably infinite number of different sizes of infinity!

    Getting back to the aleph numbers aleph i is defined as the smallest size of infinity that is strictly bigger than aleph i-1 with aleph 0 being set as countably infinite. It depends on whether you think the continuum hypothesis is true as to whether or not you think aleph i = beth i.

    Hope that was helpful and you can impress people with it next time you’re at the pub. 😉

  • Sorry, beth 2 is the size of the power set of the size of the power set of natural numbers. Wish I could edit comments here cos I spotted a few other typos. :/

  • Thanks for that. That was pretty well explained. I’ve got the idea now. Now, here’s a question. Can you give me an application of that? Somewhere where this is useful.I’m thinking back to the bit of research I did last year on what students thought of the links between physics and maths. Here, it was clear that physics students viewed maths as explaining physics, with physics meaning the real world. “Maths is simply physical things explained”, said one student. So is that the case? How are transfinite numbers be applied in the real world? (Or are they not?)

  • I’m not really aware of a direct real-world application (I say direct because I strongly suspect it will have some implications to maths that *is* used directly on real world problems). Admittedly though it isn’t my field so there might be applications but it isn’t clear to me what they might be. Moreover in practice I’ve only ever comesacross/used countably infinite and uncountably infinite (but then it’s more about describing the sorts of things methods work on—or don’t work on—rather than specifically applying it to something real world). I get the impression that it’s pretty rare to need to go beyond those two (and I note in wikipedia it only names sets of size beth 0, 1 and 2).

    But maybe find a pure mathematician and ask if they know of anything, you never know?

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