Minimal Models and β-Categoricity
Logic and Set Theory Seminar
10th March 2021, 4:00 pm – 5:30 pm
Let us say that a theory T in the language of set theory is β-consistent at α if there is a transitive model of T of height α, and let us say that it is β-categorical at α iff there is at most one transitive model of T of height α. Let us also assume, for ease of formulation, that there are arbitrarily large α such that ZFC is β-consistent at α.
The sentence V = L has the feature that ZFC + V = L is β-categorical at α, for every α. If we assume in addition that ZFC + V = L is β-consistent at α, then the uniquely determined model is Lα, and the minimal such model, Lα0 , is model of determined by the β-categorical theory ZFC + V = L + M, where M is the statement “There does not exist a transitive model of ZFC.”
It is natural to ask whether V=L is the only sentence that can be β- categorical at α; that is, whether, there can be a sentence φ such that ZFC+φ is β-categorical at α, β-consistent at α, and where the unique model is not Lα. In the early 1970s Harvey Friedman proved a partial result in this direction. For a given ordinal α, let n(α) be the next admissible ordinal above α, and, for the purposes of this discussion, let us say that an ordinal α is minimal iff a bounded subset of α appears in Ln(α) Lα. [Note that α0 is minimal (indeed a new subset of ω appears as soon as possible, namely, in a Σ_1-definable manner over L(α0+1) and an ordinal α is non-minimal iff Ln(α) satisfies that α is a cardinal.] Friedman showed that for all α which are non-minimal, V = L is the only sentence that is β-categorical at α. The question of whether this is also true for α which are minimal has remained open.
In this talk I will describe some joint work with Hugh Woodin that bears on this question. In general, when approaching a “lightface” question (such
as the one under consideration) it is easier to first address the “boldface” analogue of the question by shifting from the context of L to the context of L[x], where x is a real. In this new setting everything is relativized to the real x: For an ordinal α, we let nx(α) be the first x-admissible ordinal above α, and we say that α is x-minimal iff a bounded subset of α appears in Lnx(α)[x] \ Lα[x].
Theorem. Assume M1# exists and is fully iterable. There is a sentence φ in the language of set theory with two additional constants, ̊c and ̊d, such that for a Turing cone of x, interpreting ̊c by x, for all α
(1) if Lα[x] |= ZFC then there is an interpretation of ̊d by something in Lα[x] such that there is a β-model of ZFC+φ of height α and not equal to Lα[x], and
(2) if, in addition, α is x-minimal, then there is a unique β-model of ZFC+φ of height α and not equal to Lα[x].
The sentence φ asserts the existence of an object which is external to Lα[x] and which, in the case where α is minimal, is canonical. The object is a branch b through a certain tree in Lα[x], and the construction uses techniques from the HOD analysis of models of determinacy.
In this talk I will sketch the proof, describe some additional features of the singleton, and say a few words about why the lightface version looks difficult.