Ordered Fields and the Real Number System

The presentation introduces the concept of an ordered field, specifically how it applies to the real number system. It begins by defining what a field is—a set with two binary operations (addition and multiplication) satisfying certain axioms. It then moves on to order relations, which are sets of ordered elements satisfying axioms like the trichotomy law and the transitive law.

The first proof demonstrates that if a, b, x, y are real numbers, with a ≤ x ≤ b and a ≤ y ≤ b, then the absolute value of (x - y) is less than or equal to (b - a). This proof utilizes a lemma concerning inequalities and algebraic manipulation to show that both (x - y) and (y - x) are bounded by (b - a), thus proving the absolute value inequality.

This section outlines the axioms that define an ordered field, including the trichotomy law, transitive law, addition law for order, multiplication law for order, and the non-triviality law. It is noted that these properties alone do not uniquely characterize real numbers, as rational numbers also form an ordered field.

The second proof focuses on the triangle inequality, demonstrating that the absolute value of (a + b) is less than or equal to the absolute value of 'a' plus the absolute value of 'b'. The proof involves squaring both sides of the desired inequality and utilizing the property that the square of the absolute value of x equals x squared, along with basic algebraic steps.

The presentation then introduces the crucial axiom that distinguishes real numbers from all other ordered fields: the least upper bound property. It defines upper and lower bounds for a set 'a' within an ordered field 'R' and introduces the concepts of supremum (least upper bound) and infimum (greatest lower bound).

This proof demonstrates that if 'x' is an element of the reals and is an upper bound for a non-empty subset 'a' of the reals, and 'x' is also an element of 'a', then 'x' is the least upper bound for 'a'. This is shown by establishing that 'x' is less than or equal to all other upper bounds for 'a'.

The fourth proof is analogous to the third, showing that if 'y' is an element of the reals and is a lower bound for 'a', and 'y' is also an element of 'a', then 'y' is the greatest lower bound for 'a'. This is proven by demonstrating that 'y' is greater than or equal to all other lower bounds for 'a'.

The video concludes by explaining the significance of ordered fields and the least upper bound property. It highlights that the least upper bound property is fundamental to major theorems in analysis regarding continuous functions, derivatives, integrals, sequences, and series. This leads to the concept of completeness: an ordered field is complete if every non-empty subset with an upper bound has a least upper bound. This property uniquely characterizes the set of real numbers as the only complete ordered field, distinguishing them from rational numbers.