>>11165709This guy is on the right track. There is at least countable and uncountable infinity, and Cantor's diagonal argument is among the simplest ways of proving the difference between the two.
>>11165643To partially answer your question, OP, consider the set of all possible fractions of integer numbers, that is, the set of rationals . The set of rationals is a countably infinite set, as evidenced by the following algorithm:
Imagine the subset of all ordered pairs of natural numbers in the plane , i.e. the set . Imagine that you start at the (1,1) point and work your way "towards infinity," counting each unique point in with the natural numbers themselves. Clearly, by moving in diagonal lines and then shifting to the next diagonal once the boundaries of this quadrant of the real plane are reached, it must be possible to sequentially number every point in . Now, consider the x coordinate of this plane to be the numerator of a fraction, and y to be the denominator of that fraction. Clearly, the set contains corresponds 1-to-1 with the set of fractions with integer numerators and demoninators, which is precisely the set of positive rational numbers, and clearly, every such fraction can be numbered distinctly using the set of natural numbers. Therefore, the set of rational numbers is countable.
