Dedekind and the continuity of the real line

At the end of the 19th century, Dedekind published his solution to the "problem of the continuity of the real line". It finds an elegant way of rigorously defining the reals out of rational line and infinite subsets of the rational line (cuts). 

Read and answer the following in preparation for the class on November 1.

• Read the chapter Continuity and irrational numbers from Essays on the theory of numbers (1901) by R. Dedekind (you can download it below).
• Make sure you understand Dedekind's reasoning and write it down for yourself in current terminology.
• Does the construction require the existence of inifnite sets of rationals? If so, how does Dedekind argue for infinity?

Infinite sets were viewed as a doubtful concept by many mathematicans. Dedekind published earlier (1888) an essay on natural numbers, in an attempt to axiomatize the notion of an number.

Read and answer the following in preparation for the class on November 1.

• Read the chapter The nature and meaning of numbers from Essays on the theory of numbers (1901) by R. Dedekind (you can download it below). This is a translation of the original paper in German (1888) Was sind und was sollen die Zahlen?
• Focus on the definition of natural numbers and on the framework in which this is done: does it rely on logic (which one) or on set theory (in which formulation), or both?
• The essays sparked a well-known controversy in which an Oberlehrer Keferstein attacted Dedekind's essays. A letter exchange followed in which Dedekind attempted to clarify his position. See below Letter to Keferstein. Read it and summarize what it says.