A chapter in Countable Number Theory: the transcendence of Euler's number

Abstract

We rework Hilbert's proof of the transcendence of Euler's number \(\mathrm{e}=2.71828\dots\) so that it uses only hereditarily at most countable sets. We achieve it by using only real functions defined on sets of fractions, like \([a,b]_{\mathbb{Q}}:=\{c\in\mathbb{Q}\;|\;a\le c\le b\}\). The key tool is (Riemann) integration of real functions defined on rational intervals \([a,b]_{\mathbb{Q}}\).