Foundations of Analysis by Edmund Landau is a great little book which I’ve **mentioned before** in a couple of my posts; Landau’s book gives a detailed account of the construction of the Real Numbers starting from the Peano Axioms.

It is tempting to take the real numbers for granted to a certain extent but one of the major developments in mathematics towards the end of the 19th Century was that the real numbers can be effectively built up from five basic axioms – the axioms for the natural numbers. These axioms, in a sense, capture the most essential properties of the natural numbers – the essence of the natural numbers – starting from these five axioms and about ninety pages later we arrive at a system of numbers that coincides with our intuitive notion of the real numbers.

The video link in this post is a link to a series of lectures that I have made covering the contents of Landau’s book up to the point where the construction of the real numbers is fully complete. The basic number system is the natural numbers which is extended into the rational numbers. The irrational numbers are added before, finally, defining the real numbers. How this all happens is remarkable (although, it wasn’t without controversy originally) and through reading Landau’s book you will never ever see the real numbers in the same way again.