## Section: New Results

### Formalizing Bourbaki-style mathematics

Participant : JosÃ© Grimm.

Most of the work described here is inspired by the experiment of giving formal proofs in Coq of the exercises found in Bourbaki's exposition of set theory. However, some of the results go beyond what can be found in Bourbaki.

We implemented a paper of Sierpinski about properties of continuous ordinal functions and limits of such functions.

We implemented a paper on sums of sequences of ordinals, showing that the value obtained (which depends on the order) lies in a finite set. We also showed that this result does not hold when replacing ordinals by order types.

We implemented a paper by Tarski that says if every infinite cartinal is equal to its square, then every set can be well-ordered (this is the axiom of choice). We had to modify our library to make the use of the axiom of choice more explicit.

We continued implementing in Coq the Exercises of Set Theory of Bourbaki. We solved two of them, and proved by a counter example that three of them are false.