# Logic Seminar - Richard Matthews (Leeds)

Dates: | 23 March 2022 |
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Times: | 15:00 - 16:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Adults, Alumni, Current University students |

Title: Constructing the Constructible Universe Constructively

Abstract: Gödel's Constructible Universe, L, was introduced to show the consistency of the axiom of choice and the generalized continuum hypothesis with the axioms of ZF and is the smallest transitive model of ZF which contains all of the ordinals. In this talk we will show how to build L in the weak intuitionistic theory IKP (without infinity) using an expanded set of Gödel functions and compare the behaviour of this model to the classical version of L. We will further investigate the properties of these functions by generalising a classical result to give a sufficient condition for a transitive subclass of a model of IZF to again be a model of IZF. Importantly, in this case such a condition can be expressed by a single sentence.

We shall also explore the concept of an ordinal in the constructive setting. We shall see that, without the law of excluded middle, this concept becomes more vague and ordinals can satisfy many strange properties. In particular, we shall see that it is possible for there to be an ordinal which is not in the constructible universe, answering a question of Lubarsky.