Harvard Logic Seminar

The Harvard Logic Seminar meets Tuesdays from 5:15pm to 6:15pm in Science Center 507. Email Will Boney (wboney@math.harvard.edu) if you are interested in speaking or being added to the mailing list.

The schedule for the Logic Colloquium can be found here.

Next talk:
  1. October 17: Cameron Freer (Borelian and Remine), "Feedback Computability"
    Abstract: The notion of a feedback query is a natural generalization of choosing for an oracle the set of indices of halting computations. Notice that, in that setting, the computations being run are different from the computations in the oracle: the former can query an oracle, whereas the latter cannot. A feedback computation is one that can query an oracle, which itself contains the halting information about all feedback computations. Although this is self-referential, sense can be made of at least some such computations.
    We'll discuss feedback around Turing computability. In one direction, we examine feedback Turing machines, and show that they provide exactly hyperarithmetic computability. In the other direction, Turing computability is itself feedback primitive recursion (at least, one version thereof). We'll also briefly consider notions for parallel computation, and for Borel maps on Cantor space.
    Joint work with Nate Ackerman and Bob Lubarsky.
Future talks:
  1. October 24: TBD, "TBD"
    Abstract: TBD
  2. October 31: Sebastien Vasey, "TBD"
    Abstract: TBD
  3. November 7: TBD, "TBD"
    Abstract: TBD
  4. November 14: Linda Westrick (University of Connecticut), "TBD"
    Abstract: TBD
Past talks:
  1. October 10: Nate Ackerman, "Trees, Sheaves and Definition by Recursion"
    Abstract: We will show there is a topological space for which presheaves are the same thing as trees. We will further show that there is a sheaf on this topological space which has an important relationship with Baire space. We will then use these connections to show how a definition by transfinite recursion can be thought of as an operation on sheaves, and how the well-definedness of such a definition can be thought of as a property of the sheaf we are working on. This will then allow us to define a second order tree as a sheaf on a tree and to expand our notion of definition by transfinite recursion to all well-founded second order trees (even those which are ill-founded as normal trees). We will then mention how these techniques can be used to prove a variant of the Suslin-Kleene Separation theorem.
  2. October 3: Jason Rute, "A uniform reducibility in computably presented Polish spaces"
    Abstract: (Joint work with Tim McNicholl.) In this talk we will define a new uniform computable reducibility between computable Polish spaces. No specialized knowledge of computability theory is required. Given computably presented Polish spaces X and Y, we say x in X is reducible to y in Y if there is a Pi^0_1 subset P of Y and a computable map f : P -> X such that f(y)=x. For each space X one may consider the corresponding degree structure deg(X). For example, deg(2^omega) is (isomorphic to) the truth-table degrees, whereas both deg(omega^omega) and deg(reals) are proper extensions of deg(2^omega). This new reducibility has many motivations. First, it is based on the notion of truth-table reducibility (which we will define). Truth-table reducibility on 2^omega is too restrictive of a setting for working within Baire space or the real numbers. For example, there are functions f in omega^omega not truth-table reducible to any X in 2^omega and sequences X in 2^omega such that X/3 is not truth-table reducible to X. Our reducibility gives the correct generalization of truth-table reducibility to these spaces. Second, this project mirrors Miller's non-trivial work extending Turing reducibility to computably presented Polish spaces. Last, our reducibility grew naturally out of work of the first author on computable arcs and the second author on Schnorr randomness. For example, we show that, for the vector space R^d, every Schnorr random is found in some computable arc.
  3. September 26: Gabriel Goldberg, "The least strongly compact cardinal and the Ultrapower Axiom"
    Abstract: It is consistent that the least strongly compact cardinal is the least supercompact cardinal, but it is also consistent that the least strongly compact cardinal is the least measurable cardinal. Which is it? The Ultrapower Axiom is an abstract comparison principle motivated by inner model theory that roughly states that any pair of ultrapowers can be ultrapowered to a common ultrapower. We give a characterization of supercompact cardinals in terms of the Mitchell order and use this to prove that the least strongly compact cardinal is supercompact assuming the Ultrapower Axiom and the GCH.
  4. September 19: Will Boney, "Model-theoretic characterizations of large cardinals"
    Abstract: Compact cardinals get their names from a characterization in terms of the compactness of L_{\kappa, \kappa}. Measurable and supercompact cardinals also have characterizations in these terms, and Magidor has used second-order logic to characterize supercompacts and extendible cardinals in this way. We will continue this line of model-theoretic characterizations and discuss the characterizations of large cardinals in terms of compactness for omitting types focusing on three logics: L_{\kappa, \kappa}, second-order, and sort logic.
  5. September 12: Sebastien Vasey, "Internal sizes in $\mu$-abstract elementary classes"
    Abstract: The internal size of an object $M$ inside a given category is, roughly, the least infinite cardinal $\lambda$ such that any morphism from M into the colimit of a $\lambda^+$-directed system factors through one of the components of the system. In the category of set, the internal size of an object is its cardinality. In the category of vector spaces, the internal size is the dimension, and in the category of metric spaces, the internal size is the least cardinality of a dense subset. We will discuss questions around internal sizes in the framework of $\mu$-abstract elementary classes ($\μ$-AECs), which are, up to equivalence of categories, the same as accessible categories with all morphisms monomorphisms. We will in particular examine an example of Shelah---a certain class of sufficiently-closed constructible models of set theory---which shows that the categoricity spectrum can behave very differently depending on whether we look at categoricity in cardinalities or in internal sizes. This is joint work with Michael Lieberman and Jiří Rosický.
  6. September 5: Nate Ackerman, "Vaught's Conjecture for a Grothendieck topos"
    Abstract: In this talk we will give background on $\mathcal{L}_{\infty, \omega}(L)$, categorical logic as well as Grothendieck toposes. We will then show how to make precise a version of Vaught's conjecture for a Grothendieck topos as well as discuss various analogs of Morley's theorem which hold in all Grothendeick toposes (under mild set theoretic assumptions). If we have time we will also discuss analogs of other theorems of $\mathcal{L}_{\infty, \omega}$ for Grothendieck toposes.

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