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Mapping intervals to sets which contain them 
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#1
Sep1410, 08:36 PM

P: 83

I have recently been extremely bothered by the fact that we can construct a bijection from [0,1] onto the entire twodimensional 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 nonintuitive (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? 


#2
Sep1510, 12:39 AM

P: 534

Are you bothered by the bijection from [0,1] to [0,2] given by multiplication by 2?



#3
Sep1510, 08:48 AM

P: 810

... 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. 


#4
Sep1510, 01:19 PM

P: 83

Mapping intervals to sets which contain them
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. 


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