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Phew, you have some courage, questioning the foundations of modern Mathematics in a place like this. But I can relate to your concerns about Cantor's argument. When I first heard it, it also felt artificial and unconvincing to me. What helped me (as with many proofs and concepts in Math) was an image, a visual metaphor if you like. Imagine a very, very large paper on which you place infinitely many dots in a grid. That's the infinity you were referring to, the infinity of a for-loop, discrete infinity. Here's the trick: you can always add more dots, say by making the distance between grid points half as small, which would quadruple the number of dots in your grid. But no matter how many dots you place on the paper, ho matter how fine your grid, there will always be holes (imagine "zooming in" on a square of four grid points). In fact, most of the paper will be empty! The other kind of infinity, continuous infinity, does not have any holes. Every spot is covered. You could not add any grid point, because the whole paper itself is painted. I'm not a "full-time Mathematician", so this view may be entirely wrong. But it helped me understand and appreciate Cantor. Perhaps it did the same for you. Cheers! |
It turns out that there is a nowhere-dense set that is still uncountably infinite: https://en.wikipedia.org/wiki/Cantor_set