BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Cé\;cilia Pradic (Swansea University) DTSTART:20251016T180000Z DTEND:20251016T190000Z DTSTAMP:20260423T035932Z UID:OLS/185 DESCRIPTION:Title: Ho w unconstructive is the Cantor-Bernstein theorem?\nby Cé\;cilia Pradic (Swansea University) as part of Online logic seminar\n\n\nAbstract\ nBased on joint work with Chad Brown [1] and [2].\nThe Cantor-Bernstein th eorem states that sizes of sets can be compared meaningfully using injecti ons: if A injects into B and vice-versa\, A and B are in bijection. This i s typically proven via an explicit construction that does not involve choi ce\, but the proof cannot be constructive. For instance\, [0\,1] and (0\,1 ) can be embedded into one another but are not homeomorphic\, meaning that Cantor-Bernstein is violated in a number of models of intuitionistic set theory. Faced with this state of affairs\, we can still ask: how bad it is ?\nFirst\, we are going to see how Cantor-Bernstein implies full excluded middle. We will then turn our attention to the Myhill isomorphism theorem\ , a constructive version of Cantor-Bernstein that states that\, for any tw o subsets A\, B ⊆ ℕ that are inter-reducible via injections\, there is a bijection ℕ → ℕ that preserves them. The theorem remains true cla ssically if ℕ is replaced by an arbitrary set X\, but this is not the ca se constructively. Bauer asked if there is a nice class of sets X for whic h it does hold constructively. After checking there is no hope for this cl ass of sets to be closed under basic operations like disjoint unions\, we will see that a version of this generalized Myhill isomorphism theorem hol ds for the conatural numbers ℕ∞ by adapting the usual back-and-forth c onstruction and assuming Markov's principle. However\, this does not exten d much: this fails for 2× ℕ∞\, ℕ + ℕ∞ as well as Cantor space. We are going to see why those failures are of different flavours\, and ske tch how to make this more precise by using oracle modalities.\n\n1. Pradic \, C. and Brown\, C. E. 2022. Cantor-Bernstein implies Excluded Middle. ar Xiv preprint arXiv:1904.09193.\n\n2. Pradic\, C. 2025. The Myhill isomorph ism theorem does not generalize much. arXiv preprint arXiv:2507.05028.\n LOCATION:https://researchseminars.org/talk/OLS/185/ END:VEVENT END:VCALENDAR