R is uncountable
The content of last week’s lectures tells us that (non-empty) bounded open intervals have the same cardinality as ℝ, but not whether this cardinailty is the same as that of the set of rational numbers or integers. The question of whether the set of real numbers is countable is our next topic. We will discuss the famous “Cantor diagonal argument”, which establishes the uncountability of ℝ. This development had an enormous influence on the evolution of mathematics in the 20th century.
Here is (last year’s version of) Lecture 15.
In Lecture 16, we will discuss the Axiom of Completeness for bounded subsets of ℝ, and look at an essential difference between the set of rational numbers and the set of real numbers. If a set of real or rational numbers is bounded (above), its elements to not persist indefinitely as we move along the number line in the positive direction, we eventually pass all of them. We can consider whether the set has a greatest element or maximum. It may not, for example the open interval (0,1) does not have a maximum element. While a bounded set of real numbers may not have a maximum element, it must have a supremum or least upper bound, which is a real number. It is not true that every bounded set of rational numbers has a supremum that is a rational number. This distinction between the real and rational numbers is related to the Axiom of Completeness for the real numbers, which is our theme for Thursday’s lecture.
Here is (last year’s version of) Lecture 16.