# Revisions to enumerability and size of sets sections

As per issues 107 and 109, the material in sets-relations-functions/sizes-of-sets needs to be cleaned up. It was inconsistent in its assumption about whether functions are always total (they are, according to the definition in the preceding chapter), it gave an incorrect formal definition of enumerability (leaving out the empty set), and whenever it mentions $$\mathbb{N}$$ it inconsistently assumes that $$\mathbb{N}$$ begins at 1. Issue 109 deals with the problem that the section size-of-sets uses the cardinality notation $$\left|X\right|$$ which leads students to assume that they can manipulate cardinalities as they can in the finite case; the proposal is to replace $$\left|X\right| \le \left|Y\right|$$ with $$X \preceq Y$$ to avoid this.

Commit a6a70a4 fixes issue 109; commit 370cb02 fixes issue 107. The latter also reformulates the diagonal proofs to make them direct instead of reductio proofs.

If you teach these sections, these changes may affect you. Please comment on the issues in GitHub if you have concerns. They will be merged into the master branch in a week otherwise.