An axiomatic approach to forcing and generic extensions

Abstract

This paper provides a conceptual analysis of forcing and generic extensions. Our goal is to give general axioms for the concept of standard forcing-generic extension and to show that the usual (poset) constructions are unified and explained as realizations of this concept. According to our approach, the basic idea behind forcing and generic extensions is that the latter are uniform adjunctions which are groundcontrolled by forcing, and forcing is nothing more than that ground-control. As a result of our axiomatization of this idea, the usual definitions of forcing and genericity are derived. Résumé. Cet article présente une analyse conceptuelle du forcing et des extensions génériques. Notre objectif est de donner des axiomes généraux pour le concept d’extension forcing-générique standard, et de montrer que les constructions habituelles sont unifiées et expliquées comme étant des réalisations de ce concept. Selon notre approche, l’idée-clé sous-tendant le forcing et les extensions génériques est que ces dernières sont des adjonctions uniformes qui sont contrôlées par le forcing, ainsi le forcing n’est rien de plus que ce contrôle. Comme conséquence de notre axiomatisation de cette idée, on dérive les définitions habituelles du forcing et de la généricité. Funding. This research was partially supported by fapesp, proccess 2016/25891-3. Manuscript received 10th April 2020, revised and accepted 17th July 2020. 1. Preliminary Remarks Forcing and generic extensions are usually not given as realizations of a concept, rather they are presented as specific constructions serving a specific purpose. Indeed, there are many different constructions with the same effect and differing on technical minutiae which obfuscate its essential components. If we want to make explicit what is this specific purpose, we must first capture the general idea avoiding inessential variations. In order to accomplish that, we turn towards an axiomatic approach. The situation is analogous to that of the real number system up to isomorphism: There are many different constructions of this system, but the axiomatic ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 758 Rodrigo A. Freire approach gives us a concept behind those constructions. We wish to capture a conceptual basis for forcing and generic extensions. Our aim is to characterize forcing and generic extensions through properties (axioms) that are common to all explicit constructions of forcing predicates and generic extensions. For example, textbook definitions of forcing relation in the ground model (which is customarily denoted by ∗), generic filter, P-name and evaluation of a P-name may vary widely, but there are common properties shared by the whole variety of constructions of forcing and generic extensions. The truth lemma and the definability lemma, for instance, hold in all constructions, independently of one’s choice of basic definitions. It is important to keep in mind the analogous case of the axiomatization of the real number system. Traditional constructions of real numbers and their operations from rational numbers may vary widely, but all given constructions satisfy the characterizing axioms of complete ordered fields. We wish to achieve the same thing with our axiomatization of forcing and generic extensions. However, axioms are not chosen at random. We need a guiding idea, which can be roughly explained as follows. First of all, our strategy is to understand forcing and genericity as the main components of a single concept, the concept of forcing-generic extension. Then, to understand that a forcing-generic extension of a transitive model M given by a generic filter G is a uniform adjunction of G to M which is controlled from the ground by forcing. The notion of being groundcontrolled by forcing is made precise by the fundamental duality: M |= p φ ⇐⇒ ∀G 3 p; M [G] |=φ and M [G] |=φ ⇐⇒ ∃ p ∈G ; M |= p φ. It may be helpful to think about the above duality in informal terms, considering the slogan “generic extensions are those which are controlled from the ground by forcing, and forcing is the ground control of generic extensions”. According to our abstract account, the association of forcing and genericity is not accidental, and the conceptual core of this subject is this undissociated forcing-generic compound. The development of the axiomatic approach to forcing and generic extensions presented here parallels the exposition of the subject given in the classic paper Unramified Forcing, by Joseph Shoenfield. However, our approach is very different. Indeed, we have, in some sense, reversed the traditional approach: Axioms constitute our point of departure, and the traditional definitions of generic filters and forcing predicates in the ground model are derived from them. Most of our axioms can be found as relevant theorems in all variations of the traditional approach, such as the truth lemma, the definability lemma, the generic existence theorem, etc, and we shall briefly recall how they are proved in that approach along the way. Furthermore, Section 9 provides a construction of a standard forcing-generic extension, followed by a verification that the axioms hold in that construction, which amounts to an exposition of all those relevant theorems. Nevertheless, our axioms qua axioms are not proved in our approach. The moral of our work is that if we want the fundamental duality (axioms (7) and (8)), the uniform adjunction of G to M (axioms (5) and (6)), the generic existence (axiom (4)), and the basic properties of our control apparatus (axioms (1), (2) and (3)), then forcing and genericity must be defined in the usual way. If we also adopt the universality of P-membership (axiom (9)), then we achieve a categoricity result. Subsequently, a construction of the standard forcing-generic extension uniquely determined by the ground model and the generic filter is accomplished as a natural outcome of our development. We should prove all that, but first we must provide a framework in which the axioms can be stated. All axioms are common to all variations of the traditional approach and can be explained in simple terms, thus showing that the whole C. R. Mathématique, 2020, 358, n 6, 757-775 Rodrigo A. Freire 759 subject rests on a very general idea, instead of being a cluster of particular, ad hoc technical constructions. 2. The Notion of Forcing-Generic Framework Assume that we are given the following basic data: A transitive model M , the elements of which are called individuals and denoted by a, b, c and d , and an absolute partial order P with greatest element 1. The domain of P is an individual of M and its elements, called conditions, are denoted by p, q , r , s and t . The absolute order relation is denoted by ≤. If p ≤ q , we say that p is a condition stronger than q . Individuals can be used as parameters in formulas. Remark 1. We deal with the usual caveat about transitive models in the way Azriel Levy did in [3], which means that we work in a conservative extension of Z FC given by an additional constant M , an axiom saying that M is transitive and an axiom schema saying that it reflects every sentence of the original language. The role of the set model M is to allow generic filters, but this is not strictly necessary. Since the generic extension must be controlled from the ground, we could stay in the ground and dispense the extension as a fiction. Accordingly, we could do forcing over V, in which (i) a choice of correct conditions is given by a unary predicate symbol satisfying some axioms (see [4, p. 282]) and (ii) the statements forced from V interpret statements about a fictitious generic extension (see [4, p. 285]). In addition to our basic data, we need the control apparatus. We need to control membership in M [G] from the ground M . In order to accomplish that, M [G] must be obtained as the transitive collapse of a binary relation in M , so that we can pull-back membership in M [G] to a relation in M . This relation, given by the additional data explained in the next paragraph, is denoted by ∃ p ∈G ; M |= a ∈p b, and it means that the collapsed a is an element of the collapsed b according to a correct condition. Therefore, we need a ternary relation R in M which we require to be definable and absolute for transitive models. This relation is called the P-membership relation and its satisfaction by the triple (p, a,b) is represented by M |= a ∈p b. Roughly, this is a “membership according to p” relation, and it is the first step towards a control apparatus. However, since the corresponding collapse fails to be injective, we shall adjust our membership control through the forcing predicates. As we have just mentioned, the control apparatus needs to be refined and completed, which gives rise to the forcing predicates. They are intended to be the ultimate control apparatus, given as follows. For each formulaφwith n free variables, a definable n+1-ary predicate φ in M called the forcing predicate corresponding toφ. The satisfaction of the predicate φ by the condition p and the n-tuple of individuals a is represented by M |= (p φ)[a]. These predicates constitute the ultimate control of M [G]. If p is a correct condition, that is one which is in G , and M |= (p φ)[a], then the formula φ is satisfied by the corresponding sequence of elements in M [G]. The final ingredient of our framework is the genericity property C . We think of a set satisfying C as a subset of P embodying a choice of conditions which are then considered to be correct. These sets are required to be filters of P which are called generic and denoted by G and H . Why are sets determined by conditions which are subsequently considered to be correct required to form filters? Because (i) if p is considered to be a correct condition and q is weaker than p, then q must be considered correct since it is “contained in p”, and (ii) if p and q are considered to be correct condi