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    • Theorem 3

    or

    Infinity

    Actually, this is a true theorem. The proof I provided is valid. What make this even more amazing is that fractions, which we also call rational numbers (read about this on the Analysis page), are dense in the real numbers. This means that between any two real numbers, you can find a rational number (can you prove this?) Compare this to the non-negative integers, which are certainly not dense in the reals.

    What we're really talking about here is infinity. There are certainly infinitely many non-negative integers (can you prove this?), just as there are infinitely many rational numbers, and real numbers.

    But there are many more real numbers than there are rationals (can you prove this? Do you need a hint?) We have just shown that there are the same number of rationals as there are integers (why can I leave out non-negative?)

    In other words, there are different kinds of infinity. We say that the integers (and hence the rationals) are countably infinite, while the reals are uncountably infinite.

    Think about the following sets of numbers and decide if they are countably or uncountably infinite, or something else. Can you prove your answers?

    • prime numbers (can you prove these are infinite?)
    • irrational numbers
    • all numbers between 0 and 1
    • all points on the number line
    • all points in the plane
    • all points in 3-dimensional space
    Keep in mind: if something is countably infinite there is a one-to-one correspondence between it and the non-negative integers; if something is uncountably infinite there is a one-to-one correspondence between it and the real numbers. This means that there is a one-to-one correspondence between any two countably infinite sets, and a one-to-one correspondence between any two uncountably infinite sets. How does that make you feel about your answers?

    One thing to remember: infinity is not a number, and doesn't have to behave like one! The mathematician most responsible for developing our understanding of infinity is Georg Cantor. You can read more about Cantor in the MacTutor History of Mathematics Archive.