Essays on the Theory of Numbers by Richard Dedekind

 

Dedekind's definition of the real numbers in the 19th century has become mathematical cannon. It is interesting to hear in his own words how he regards the continuum and infinity.

Dedekind defines real numbers as certain sets of rational numbers. As you may recall, set theory wasn't formally axiomatized until the 20th century. So, the set theory Dedekind was working in was a naive set theory–and one that was shown to have serious problems by Bertrand Russell (see Russell's Paradox).

Yet, Dedekind's definition of the reals remainder popular enough to be recharacterized in Zermelo-Fraenkel set theory (ZF) and (presumably) all of the other proposed set theories. 

I have some thoughts on the reals and how something weird is going on with them. Part of the "blame" lies with Dedekind. Keep an eye out for a post about this shortly!