View: session overviewtalk overviewside by side with other conferences

09:45 | Infinitely deep languages and neighbors SPEAKER: Tapani Hyttinen ABSTRACT. We take a look on the development on the questions around infinitely deep languages that has happened during the last 40 years. |

11:15 | Computing a floor function SPEAKER: Julia Knight ABSTRACT. For the field of real numbers, we have the usual floor function, with range equal to the set of integers. If we expand the reals, adding the function 2^x, then for a positive integer x, 2^x is also a positive integer. Mourgues and Ressayre showed that every real closed field has an ``integer part''. The construction is complicated, not arithmetical, but hyperarithmetical. Ressayre showed that every real closed exponential field has an ``exponential integer part''. This construction is even more complicated. It may not finish in the least admissible set over the input. |

15:15 | Functors in Computable Model Theory SPEAKER: Russell Miller ABSTRACT. We give an overview of a number of recent results in computable model theory, by various researchers (not necessarily including the speaker). The results are not all directly connected to each other, but they serve to illustrate the principle that much of the work in this discipline can be viewed through the prism of functors, on categories C and D whose elements are countable (or computable) structures and whose morphisms are isomorphisms (not necessarily computable). Ideally, such a functor F from C to D should be effective: given a structure M from C as an oracle, it should compute the structure F(M) in D, and given a C-morphism g from M to N as an oracle, it should compute the D-morphism F(g) from F(M) to F(N). Moreover, one would hope for F to be full and faithful, as a functor, and to have a computable inverse functor. In practice, it is unusual for an F to have all of these properties, and for particular applications in computable model theory, only certain of the properties are needed. Many familiar examples will be included to help make these concepts clear. |

16:45 | Classes of structures with no intermediate isomorphism problems SPEAKER: Antonio Montalban ABSTRACT. We say that a theory $T$ is intermediate under effective reducibility if the isomorphism problems among its computable models is neither hyperarithmetic nor on top under effective reducibility. We exhibit partial results towards showing that the existence of such theories is equivalent to the negation of Vaught's conjecture. We prove that if an infinitary sentence $T$ is uniformly effectively dense, a property we define in the paper, then no extension of it is intermediate, at least when relativized to every oracle in a cone. As an application we show that no infinitary sentence whose models are all linear orderings is intermediate under effective reducibility relative to every oracle in a cone. |