Can there be natural instances of nonlinearity in the hierarchy of consistency strength? UWM Logic Seminar, January 2021

This is a talk for the University of Wisconsin, Madison Logic Seminar, 25 January 2020 1 pm (7 pm UK).

The talk will be held online via Zoom ID: 998 6013 7362.

Abstract. It is a mystery often mentioned in the foundations of mathematics that our best and strongest mathematical theories seem to be linearly ordered and indeed well-ordered by consistency strength. Given any two of the familiar large cardinal hypotheses, for example, generally one of them proves the consistency of the other. Why should this be? The phenomenon is seen as significant for the philosophy of mathematics, perhaps pointing us toward the ultimately correct mathematical theories. And yet, we know as a purely formal matter that the hierarchy of consistency strength is not well-ordered. It is ill-founded, densely ordered, and nonlinear. The statements usually used to illustrate these features are often dismissed as unnatural or as Gödelian trickery. In this talk, I aim to overcome that criticism—as well as I am able to—by presenting a variety of natural hypotheses that reveal ill-foundedness in consistency strength, density in the hierarchy of consistency strength, and incomparability in consistency strength.

The talk should be generally accessible to university logic students, requiring little beyond familiarity with the incompleteness theorem and some elementary ideas from computability theory.

4 thoughts on “Can there be natural instances of nonlinearity in the hierarchy of consistency strength? UWM Logic Seminar, January 2021

  1. Wonderful! I can’t wait. I was precisely reading Maddy on Steel’s Multiverse (in particular, in comparison with yours), when notified of this event.

  2. My impatience was rewarded beyond my expectations! Bravo again. Sorry, I feel like a spamming groupie, but hey, your work only has not to be so exciting so that I stop.
    Just wanted to ask you something I did not have time to yesterday: would you have any idea of the order type of the consistency strength order (if ever there are canonically regstered ill-founded & nonlinear order types…)?

    • In my paper, which we shall be reading in the seminar in which I believe you are participating this term, I prove that this hierarchy contains copies of the free countable Boolean algebra, and so in particular it is universal for all countable partial orders. So it is very complicated.

      • Oh great! Your belief is right. Thank you!
        Okay, such as the Turing Degrees then… Makes me wonder whatis known about universal countable partial orders, more than mutual embeddability… But those two instances plus K.J. Williams’ T-realizations of any countable model of ZFC, already prompts to think they’re even more complicated!

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