## Section: New Results

### Formalization of Bourbaki's sets and ordinals

Participant : José Grimm.

In previous years we developed a formal library describing the parts of the Bourbaki books on set theory, cardinals and ordinals. We completed it by adding the definition of real numbers using Dedekind cuts. The important properties we showed that $\mathbf{R}$ is an ordered Archimedean field, that every non-empty bounded subset has a least upper bound, that every Cauchy sequence has a limit, and that the intermediate value theorem holds.

It follows that every positive real number has positive square root. We give a pair of adjacent sequences that converges to this square root. For instance $\sqrt{2}$ is irrational, and we get a pair of rational adjacent sequences that converges to it. This produces an explicit order isomorphism ${\mathbf{Q}}^{*}\to \mathbf{Q}$. The number of such isomorphisms is equal to the power of the continuum (the cardinal of $\mathbf{R}$) [18] .