In emanuela's post above, about the real numbers being of a higher order of infinity than the integers, it should be noted that the numbers represented by, e.g., 2.999999999... do not exhaustively represent the real numbers. A rational number is one that can be represented as the ratio of two integers. Thus, 0.3333333... can be represented as the ratio of 1 to 3. A feature of the rational numbers is that they can be represented as a repeating decimal, i.e., one in which a pattern of some finite number of digits repeats indefinitely at the end. 1/11=0.090909..., for example and 1/13=0.076923076923076923.... In some cases the repeating digit(s) will be 0, e.g., 1/8=0.12500000... In this case there is a non-repeating string of digits before we get to the repeating part. The repeating part of one of this type doesn't have to be zero; 2/15=0.13333333...

There is another type of real number, the irrational. Examples are pi, the square root of two, and e, the base of the natural logs. These cannot be represented as repeating decimals.

The the order of infinity of the rational numbers is the same as the order of infinity of the integers. Bean or emanuela could prove this for us. Add in the irrational numbers and *that's when you get the higher order of infinity.

This notwithstanding the fact that, as Alex pointed out, between any two rational numbers you can put more rational numbers, even an infinite number of rational numbers; there are no adjacent pairs of rational numbers.