|The reverse mathematics of Hindman's theorem|
du groupe Séminaire Méthodes Formelles
|Date|| 2020-02-11 07:00-07:00|
|Titre||The reverse mathematics of Hindman's theorem |
|Résumé||Reverse mathematics consists of the study of the minimal axioms needed to prove a theorem. A phenomenon that appeared at the beginning of this study is that among the "natural" theorems, an enormous majority are equivalent to one out of five axiomatic systems: the Big Five. The exceptions to this are mainly mainly theorems from combinatorics, the most famous being the Ramsey theorem for pairs. This makes combinatorics especially interesting in the context of reverse mathematics.
Some theorems still escape classification, such as Hindman's theorem, which says that for any finite coloring, there is an infinite set whose set of finite sums of distinct elements is monochromatic. This theorem has several proofs, but the most elementary is in the system ACA_0^+ (the existence of the omega-jump), while it is only known to imply arithmetical comprehension.
We will start by presenting reverse mathematics and its use of calculability, before applying it to Hindman's theorem, presenting a sketch of the most elementary proof, and the latest advances on this subject. |
|Orateur||Paul-Elliot Anglès Auriac |
|Url||Institut Camille Jordan, Lyon |
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