Reverse mathematics and the Ramsey theorem for pairs
Logic and Set Theory Seminar
12th February 2020, 3:00 pm – 4:30 pm
Fry Building, G.07
Every mathematician already heard his colleagues of professors say things like ``Theorems A and B are equivalent'' or ``Theorem C does not follow from theorem D''. But what do we really mean by that ? Indeed, from the viewpoint of classical logic, theorems are all pairwise equivalent within ZFC. Also even though it is formally true, no one would claim that the four color theorem implies the Fermat-Wiles theorem. Reverse mathematics can be seen as an attempt to put a formal meaning on our intuition about this : two theorems are equivalent if each can be deduced from the another one, using in addition only a restricted set of axioms: the ones allowing to perform "elementary transformations", that is, simply computations. The terminology itself comes from the used method to show axioms optimality: Once some axioms have been used to show a theorem T, we then try to demonstrate back these axioms from Theorem T, with the only help of the "ground axioms" allowing to perform elementary transformations : this is why these mathematics are "reversed".
A simple observation that certainly contributed to the growth of reverse mathematics, is that they are very structured : Most mathematical theorems are equivalent to one among five axiomatic systems linearly ordered by strength: RCA0, WKL0, ACA0, ATR0 and Pi11-CA0. Among them RCA0 is the weakest which is also the one we always take for granted : it contains the basic axioms, needed to develop the computable mathematics. The systematic equivalences of theorems with one of these five systems led to them sharing the nickname "Big Five".
One of the most studied theorem in reverse mathemetics is the Ramsey theorem for pairs - RT22 - which is the cornerstone of a 50 year mathematical adventure. We will try to explain what is special about this theorem, by going through the history of its related reverse mathematical results, from the non-provability of RT22 within RCA0, to the separation between RT22 and SRT22 in omega-models.