Note that there is more than one way of discussing how big an infinite set of number is. The approach described in this thread, by posters such as Chronos and ITR champion, is the set theory approach, which deals with cardinality (the number of individual members a set has). (By the way, here’s a good beginner-level explanation I’ve linked to before.)
But you can also talk in terms of measure theory, in which the measure of something like an interval of real numbers is just its length, and so the set of real numbers between 0 and 2 is, in this sense, twice as “big” as the set of real numbers between 0 and 1. Things get complicated and subtle, though, when you start to measure sets that aren’t a single, connected interval, but are made up of (infinitely many) separate pieces.
And you can’t really find the sum of infinitely many numbers, but you can find their average (as in using a definite integral to find the average value of a function over an interval).
No, it’s mathematics and intuition that part ways. Logic is still very much in effect, as all of these counterintuitive results follow logically from the definitions you start with.
+1. Following the established rules to their logical conclusion can often lead to unintuitive results. People rejected Cantor’s work for a long time because his ideas about cardinality seemed so counterintuitive. (There are the same number of integers as even integers? Ridiculous!)
Nitpick: While Cantor’s work was ridiculed by many serious mathematicians, the same-sizedness of the integers and even integers was not the cause of their ridicule. That “paradox” was already known to Galileo, Archimedes and others and was not controversial. Nor were proofs of key early theorems (e.g. that integers and rationals are equal-sized, while integers and irrationals are not) in contention. The objections raised by Kronecker and others were instead based on the alleged uselessness and non-reality of Cantor’s set theory.
It’s not just following rules, it is establishing useful definitions, too. The idea that you can define the idea of two sets having the same size, without first having a number to represent that size, is new and interesting and allowed us to ask questions like, “do all infinite sets have the same size?”
I didn’t mean logic as in ‘formal logic’ but as in … err … informal logic?
In fact, I wrote ‘common sense’ at first but somehow felt that wasn’t the right word. It’s these strange moments that I realise what happens when writing in a language that I sort of master, but still is not my first …
Mind-blowing fact: there are no more points in a square-inch than in a line the length of an inch.
Every point in [0 , 1) × [0 , 1) can be represented in decimal notation as 0.xyxyxy…, where the x:s denote the successive decimal places of the first coordinate and the y:s those of the second.
Got it? I didn’t really believe it at first!
There are an equal number of real numbers between any pair of real numbers x and y, as long as x is less than y. (To be more strict: The cardinality of the set of real numbers in the open interval (x, y) is aleph-one, assuming x < y.) It doesn’t matter how close x and y are.
The proof is actually pretty easy to see. Imagine the graph of the tangent function. Between any pair of asymptotes in that graph, the function maps every value in that region of the x axis to a unique value on the y axis (more precisely: in every open interval on which some function tan(a*x)+b is everywhere differentiable, it is a bijection, for a and b real numbers, a not zero), which proves that the given region on the x axis has as many real numbers as the entire y axis, which is, of course, the entire real line.
Since we can make the period of the tangent function arbitrarily large, we can make the intervals between asymptotes arbitrarily small, and the same property holds. Therefore, every non-empty open interval on the real line contains as many real numbers as the entire real line.
Are you sure it has to be finite? Suppose I have a point defined by a (countably) infinite number of coordinates. I put them in a list (like Cantor’s diagonal). My single real coordinate is generated by starting at the upper left, then traversing in a zig-zag pattern towards the lower-right, covering every possible digit.
In order to construct the diagonalisation you need to show that you can complete even one diagonal. You have two countably infinite things going on here. I don’t think you can do it. Would need to think a bit more, but I have always assumed this is the point where it breaks.
Having two countably infinite things isn’t a problem–the enumeration of the rationals proceeds in a similar way, and I know they’re countably infinite (make a grid of fractions with numerators along X and denominators along Y, then enumerate in a zig-zag, skipping unreduced fractions).
As you know, while the set of integers is infinite, every individual element is finite. Likewise (I think!), while the set of coordinates may be infinite, the index of every particular coordinate is finite. Same goes for the index of every digit in that coordinate. For any finite (coordinate index, digit index) combo, I can always give you the index into the constructed coordinate, even with an infinite number of input coordinates (something that you cannot do with the usual interleaving trick).
That said, I may well have screwed up somewhere. But it seems possible based on my understanding.
I think the answer is it depends how infinite and also what definition of dimension is employed.
The cardinality of an infinite dimensional real vector space V will be the cardinality of the continuum if the dimension (which is itself the cardinality of the maximal linearly-independent subset) is less than or equal to the cardinality of the continuum.
However (in physics at least) when you have infinite dimensional vector spaces the vector space dimension is not very useful and the dimensionality is usually defined using some additional structure on the vector space. Specifically the Hilbert space dimension is used, which is the cardinality of an orthonormal basis. I believe that the cardinality of an infinite dimensional (real) Hilbert space is still the cardinality of the continuum if the (Hilbert space) dimension is less than or equal to the cardinality of the continuum, though this may require additional assumptions (like the continuum hypothesis). That seems like something Indistinguishable would probably know!
