>>14384738It does sound like there should be more numbers than but there is actually an equal number of them. This is because rational numbers can be enumerated using the "zig-zag" pattern (1/1, 1/2, 2/1, 3/1, 2/2, 1/3,...). If you can enumerate something it means you can say "this is first, second, third" (1, 2, 3...) which means you made a map where each natural number corresponds to a rational number. If you can make such a map between two sets (i.e. a bijection) it means those two sets have an equal number of elements. In addition, the number of integers is equal to the number of natural numbers because they can also be enumerated (0, 1, -1, 2, -2, 3, -3...). To summarize:
On the other hand, there are more real numbers than natural numbers because of "Cantor's diagonal argument". Even if you enumerate all the real numbers you can with a map you can still construct another real number that was not counted, by picking a unique decimal from each enumerated real number. Similar to how unintuitively there is an equal number of rational numbers and natural numbers, there is an equal number of complex numbers and real numbers. There is also an equal number of real numbers on an interval (which is a subset of ) as there are on entire and there is an equal number of points in 3D space as there are on a 1D line . To summarize:
>*Norman J. Wildberger enters the scene.*This is all bullshit because infinity is bullshit.
