Forcing-theoretic framework for the Fraïssé theory

The abstract of the dissertation Forcing-theoretic framework for the Fraïssé theory Ziemowit Kostana 15.03.2021 Subject of the dissertation The subject of the dissertation is the study of possible generalizations of the Fraïssé theory, using the method of forcing. In 1954 Roland Fraïssé discovered that many classes of finite models, like graphs or linear orders, can be canonically assigned certain infinite models. These infinite models are universal – they contain isomorphic copies of all finite models from the class – and homogeneous – each isomorphism between finite substructures can be extended to an automorphism of the whole structure. An infinite structure with these two properties is called a Fraïssé limit. This correspondence is reversible – given a countable, homogeneous model, one can recover the class of finite models from which it was built – it is exactly the class of its finite substructures. The Fraïssé theory studies this correspondence. The Fraïssé limit of a class K has a natural connection with the forcing Fn (!;K; !) = fA 2 Kj F(A) 2 [!]<!g; where F(A) denotes the universe of a structure A, and the ordering is given by the reversed inclusion of substructures. If G Fn (!;K; !) is a filter intersecting sufficiently many dense sets, then S G is a structure isomorphic to the Fraïssé limit of K. This inspires a natural question about structures added in a similar way by the forcings Fn (S;K; !) = fA 2 Kj F(A) 2 [S]<!g; for uncountable sets S, in particular S = !1. The study of such structures is the topic of Chapters 4 and 5. It should be emphasized, that despite the obvious model-theoretic aspect, this is a dissertation about the set theory. The apparatus of model theory is very basic. On the other hand, the set-theoretic machinery is rather sophisticated, and this refers particularly to forcing-theoretic arguments. Although almost all forcing notions appearing in the dissertation are c.c.c. (and indeed most of them resemble the Cohen forcing) arguments sometimes get quite technical and involved. Chapter 1 The chapter is a brief survey of the development of the Fraïssé theory, its relatives studied in the past, and their applications. Chapter 2 The chapter is an introduction to the classical Fraïssé theory, with examples. WE describe the Fraïssé limits of the following classes: linear orders, graphs, Boolean algebras, partial orders, and metric spaces with rational distances. Chapter 3 We introduce the Fraïssé-Jónsson theory, which is a modification of the classical Fraïssé theory, where we do not assume that the models from K are finite. Examples of the uncountable Fraïssé limits we can obtain this way, are countably saturated models of size 2!, in case 1 when 2! = !1. Some of such models admit natural representations, like the Boolean algebra P(!)=Fin. The Fraïssé-Jónsson theory essentially uses assumptions on the cardinal arithmetic, so it is natural to look for some canonical, saturated models, whose existence is not dependent on additional axioms of set theory. As an illustration, let us note that without CH it might not be true that a countably saturated linear order of size 2! is unique. However, among such orders there is always a unique "minimal" one – namely the one which can be isomorphically embedded into any countably saturated linear order. Chapter 4 We study structures added by the forcings fA 2 Kj F(A) 2 [S]< g; where is an infinite cardinal (typically = !), S is any set, F(A) stands for the universe of a structure A, and the ordering is given by the reversed inclusion of substructures. If jSj = !, the generic structure is the Fraïssé limit of the class K (under some reasonable assumptions on K), while if jSj > !, the generic structure is typically rigid, i.e. has no automorphisms other than the identity. By slightly modifying the forcing, we can add structures together with automorphisms. Recall, that a linear order is !1-dense if each nonempty open subset has size !1, and separable if has a countable dense subset. Theorem (Kostana). The following is consistent with ZFC: There exists an !1-dense, separable linear order (A; ) together with an automorphism : A ! A, such that • 8a 2 A (a) > a; • Aut (A; ) = f kj k 2 Zg: In particular Aut (A; ) ' (Z; +). Chapter 5 We continue the study of models from Chapter 4, focusing on the models of size !1. Applying the ideas of Avraham, Rubin, and Shelah from the 80s, we show that in many cases the generically added structures become homogeneous after suitably extending the model of set theory, so that it satisfies Martin’s Axiom. In this extended models, they also have certain Ramsey-like properties. Theorem (Avraham-Shelah). The following is consistent with ZFC +MA!1 : There exists an !1-dense separable linear order L with the property that each uncountable partial function f L L is monotone on an uncountable set. Theorem (Kostana). The following is consistent with ZFC +MA!1 : There exists a separable metric space X of size !1 with the property that each uncountable partial 1-1 function f X X is an isometry on an uncountable set. Moreover, distances between points of the space X are rational, and X has a dense copy of the rational Urysohn space. 2