OK… I think that this deserves a new thread from the probability post. Physicists and mathematicians alike feel free to respond to this.
Is infinity an absolute, or merely a unit of convenience for equations? What if any errors are assumed when using infinities in theories/proofs/etc?
I’d like to offer a speciifc example, but there are so many out there, and I think it might be good if you could illustrate your point by using an example you know already.
Infinites bug me since I cant get around the concept of something like 1/infinity = 0 thoguh I am grudgingly willing to accept that it is 0 and not just really excruciatingly tiny.
thanks for all the light that will be shed
Infinity in physics is merely a way of saying, “this function (be it a wave or a field)” goes out a really far way, so far, you might as well say it’s infinity. There are quite a few considerations you have to make for functions that, even theoretically, go out to infinity. Most of the time we demand that they go to zero (otherwise we’ll violate some conservation law), or, more rigorously, that the normalized function has total probability of one (the integral taken out to infinity is unity). An infinite plane wave or an infinite wavefunction is of some convenience in theoretical treatments. In reality there’s no such thing as an infinite plane wave.
Ironic that thanked us for light that will be shed, as light happens to be one of the phenomenon that infinite plane waves are considered.
See this thread, or do a search for infinity in GQ for lots of information.
In summary, Infinity is a concept, not a number (otherwise you could add 1 to it and have a number greater than infinity).
There are many different concepts of infinity. There is the infinity of infinite series, e.g. the sum from 1 to infinity of 1/2^n is 1. This has a precise meaning that can be given in purely finite terms, but I won’t bother since it can be found in standard places. Then there are the cardinalities of infinite sets. For example, countable infinity is the cardinality of the integers. The continuum is the infinity of the points on the line. (The assumption that there is no infinity is called the continuum hypothesis and is undecidable in the usual axioms of sets.) There are further infinities of that sort. On the other hand, the belief that 1/0 = infinity is nonesense, since there is no way of reversing division by 0. So infinity is not a number. On the other, among Conway’s surreal numbers, there are many infinite numbers that do allow arithmetic. But their inverses are not 0; rather they are infinitesimals. Which is another kettle of fish. Read Knuth’s wonderful book on the subject.