I have recently been extremely bothered by the fact that we can construct a bijection from [0,1] onto the entire two-dimensional plane which itself contains [0,1]. Similarly, I have been bothered by the fact that we can construct a bijection from (0,1) to all real numbers. Indeed we do so routinely with functions like [tex] f(x) = tan\left(\pi x - \frac{\pi}{2}\right).[/tex] Referring back again to my first example (that is, about [0,1] and the plane): every single point that is in [0,1] is also in the two dimensional plane. It just seems absurd that you can map [0,1], to put it crudely, to itself and more (the rest of the plane). I have known for a while that we are able to do things like this and I previously simply regarded it as a rather surprising and non-intuitive (though equally interesting) result. However, while recently thinking and studying more on the structure of the real numbers, these mappings just seem rather absurd...not just surprising, but absurd in the sense that I am honestly considering abandoning study of real analysis all together. Things like this seem to suggest that there is something...metaphysically...wrong with the whole system. Why, then should I study it? I have long considered real analysis to be one of the most beautiful areas of mathematics. So, in essence I am asking someone to dig me out of this, to preserve the beauty for me, to convince me that I can remain in this world without disconnecting myself totally from reality. I am sure previous mathematicians have struggled with this. Any suggestions?
... or the bijection between N and Z+ given by adding 1? Note that the bijection from R to R^2 is not a particularly nice one. It's not continuous. It might better settle your mind to think of it as "encoding" two real numbers in a single real number. For example, if I have two real numbers a and b such that a = Σa_n * 10^n and b = Σb_n * 10^n, I can create a real number c = Σa_n * 10^(2n) + b_n * 10^(2n+1). With this, we have defined a function f(a, b) = c, which is your bijection.
Yes, I am bothered by this as well. Are others not also bothered? If you think about it as a stretch of a continuous elastic substance, then it is not bothersome. But, in modern analysis we act as though the interval [0,1] is a set composed of points and we say that [0,2] is also a set composed of points which contains all of the points in [0,1] and more. If we accept this idea of the intervals being composed of points, then a bijection between the two seems problematic. It is really the fact that the formalism of analysis speaks of these intervals as sets that is highly bothersome to me. This one bothers me, but definately not as much as the ones with the real numbers. And the reason it fails to irk me as much may be attributable to the fact that I might be thinking of the integers as an 'uncompleted' infinity (i.e. not really a completed set, but a set which is never fully formed), whereas I think of intervals on the real number line as fully completed because we imagine them to represent portions of (a conceptual) space. And it would seem that space (at least in our conceptual/ideal world) does exist as a completed reality. The idea of encoding two reals into another real is indeed an interesting one. I appreciate that input, I will give it some more thought. That said, however, the problem still remains if we want to think about continuous space (for simplicity lets just consider the 1 dimensional continuum...a line) as consisting of infinitely many points which are labeled by R. It seems to me that if we wish to speak about portions of lines as being composed of sets (for instance, if we think of the intervals [0,1] or [0,2] as sets composed of objects), then we end up with results that do not make much sense.