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].(adsbygoogle = window.adsbygoogle || []).push({});

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 itselfand 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?

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# Mapping intervals to sets which contain them

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