Next week Giovanni Solda is visiting us from Ghent. He will give a talk on "A combinatorial principle weak over weak systems yet strong over strong systems".
Abstract:
Better quasi orders (henceforth bqos) are a strengthening of the notion of well quasi order. Even if their definition is more complicated, the former enjoy nice closure properties that make them, in a way, easier to work with than the latter: this feature made bqos an instrumental tool in proving landmark results like Nash-Williams' theorem and Laver's theorem. From the reverse mathematical point of view, the study of bqos is an interesting area still full of open questions.
In this talk, we will focus on a property of non-bqos, the so-called minimal bad array lemma, and in particular one version of it that we will call MBA − . We will show that MBA − has a very odd behavior when it comes to its reverse-mathematical strength, namely
• over ATR 0 , MBA − can be seen to be equivalent to the very strong principle of Π 12 -comprehension, yet
• over ACA 0 , MBA − does not imply ATR 0.
In order to prove this result, we will provide a characterization of the quasi-orders that ACA 0 proves to be bqos.
This is joint work with Anton Freund, Alberto Marcone, and Fedor Pakhomov.
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