Foundations of Finitely Supported Structures: A Set Theoretical Viewpoint

We introduce the theory of atomic finitely supported algebraic structures (that are finitely supported sets equipped with finitely supported internal operations or with finitely supported relations), and describe topics related to this theory such as permutation models of Zermelo-Fraenkel set theory with atoms, FraenkelMostowski set theory, the theory of nominal sets, the theory of orbit-finite sets, and the theory of admissible sets. The motivation for developing such a theory comes from both experimental sciences (by modelling infinite algebraic structures hierarchically defined by involving some basic elements called atoms in a finitary manner, by analyzing their finite supports) and computer science (where finitely supported sets are used in various areas such as semantics, domain theory, automata theory and software verification). We describe the methods of translating the results from the non-atomic framework of Zermelo-Fraenkel sets into the atomic framework of sets with finite supports, focusing on the S-finite support principle and on the constructive method of defining supports. We also emphasize the limits of translating non-atomic results into an atomic set theory by presenting examples of valid Zermelo-Fraenkel results that cannot be formulated using atomic sets. 1.1 A Short Introduction There does not exist a formal description of infinity in those sciences which are focused on quantitative aspects. Questions such as ‘What represents the infinite?’, ‘How could the infinite be modelled?’, or ‘Does the infinite really exist or is it just a convention?’ naturally appear. In order to provide appropriate answers, we present ‘finitely supported mathematics’ (FSM) which is a generic name for ‘the theory of finitely supported algebraic structures’. FSM is developed by employing the general principle of finite support claiming that any infinite structure hierarchically defined by involving some basic elements called atoms must have a finite support under a canonical permutation action. Informally, in FSM framework we can model infinite structures (defined by involving atoms) by using only a finite number of character1 © Springer Nature Switzerland AG 2020 A. Alexandru, G. Ciobanu, Foundations of Finitely Supported Structures, https://doi.org/10.1007/978-3-030-52962-8_1 2 1 The World of Structures with Finite Supports istics. More precisely, in FSM we admit the existence of infinite atomic structures, but for such an infinite structure (hierarchically constructed from / 0 and from the set of atoms) we consider that only a finite family of its elements (i.e., its ‘finite support’) is ‘really important’ in order to characterize the related structure, while the other elements are somehow ‘similar’. As an intuitive motivation in a λ -calculus interpretation, the finite support of a λ -term modulo α-equivalence is represented by the set of all ‘free variables’ of the term; these variables are those who are really important in order to characterize the term, while the other variables can be renamed (by choosing new names from an infinite family of names) without affecting the essential properties of the λ -term. This means that we can obtain an infinite family of terms starting from an original one (by renaming its bound variables), but in order to characterize this infinite family of terms it is sufficient to analyze the finite set of free variables of the original term. Finitely supported mathematics has connections with the Fraenkel-Mostowski permutation model of Zermelo-Fraenkel set theory with atoms [41], with FraenkelMostowski axiomatic set theory [29], with the theory of nominal sets on countable sets of atoms which do not have an internal structure [44], and with the theory of generalized nominal sets on sets of atoms which may have an internal structure [23]. Actually, FSM corresponds to Pitts’ nominal sets theory by analyzing nominal sets (or, more generally, finitely supported subsets of nominal sets) endowed with a finitely supported algebraic structure (such as nominal monoids, nominal groups, nominal partially ordered sets, etc) and with the mention that the countability of the set of basic elements is ignored. Nominal sets are called in this book ‘invariant sets’, motivated by Tarski’s approach on logicality; this aspect is explained below. Intuitively, FSM is the algebraic theory obtained by replacing ‘object’ with ‘finitely supported object (under a canonical permutation action)’ in the classical Zermelo-Fraenkel set theory (ZF). The principles of constructing FSM have historical roots both in the definition of logical notions by Alfred Tarski [49] and in the Erlangen Program of Felix Klein for the classification of various geometries according to invariants under suitable groups of transformations [38]. There also exist several similarities between FSM, admissible sets introduced by Barwise [19] and Gandy machines used for describing computability [30]. FSM sets are finitely supported subsets of invariant sets (where invariant sets developed over countable families of atoms are actually nominal sets). We use a slightly different terminology motivated by Tarski’s approach regarding logicality (i.e. a logical notion is defined by Tarski as one that is invariant under the permutations of the universe of discourse) and because our results are related to foundations of mathematics, i.e. we study choice principles, results regarding cardinality order, results regarding cardinality arithmetic, results regarding various forms of infinity (Dedekind infinity, Tarski infinity, Mostowski infinity, ascending infinity, etc), results regarding fixed points, results regarding the connections between atomic and non-atomic sets, results regarding finitely supported binary relations and so on. The value of the nominal approach is recognized. However, it is related to computer science applications (which are described well in [44]), while we work on foundations of mathematics by studying the validity, the consistency and the inconsistency of various results in 1.1 A Short Introduction 3 an ‘atomic’ refinement of ZF set theory. Actually, our work is connected also with the Fraenkel-Mostowski set theory (FM), admissible sets and amorphous sets (that inspired actually the ‘nominal approach’). We do not minimize the benefits of nominal approach that has significant applications in areas such as semantics, automata theory and verification, but regarding the foundations of finitely supported structures we consider that ‘FM’, ‘invariant’ or ‘FSM’ are more adequate names. Our book can be seen as an in-depth study on the foundations of FM sets and nominal sets (defined over families of basic elements that are not necessarily countable) accessible even to graduate students. For computer science applications of nominal sets (that are not treated in this book) and for generalizations in this direction (orbit-finite sets, automata over infinite languages etc), we strongly recommend the book [44]. The original motivation for developing finitely supported sets has its roots in the study of the independence of the Axiom of Choice (AC) claiming that for any family of nonempty sets F there exists a set containing exactly a single element from each member of F . Since its first formulation AC has conduced to several debates and controversies. The first controversy is about the meaning of the word ‘exists’ since this term is very ‘abstract’. One group of mathematicians (called intuitionists) believes that a set exists only if we are able to provide a method of constructing it. Another controversy is represented by a geometrical consequence of AC known as the Banach and Tarski’s paradoxical decomposition of the sphere which shows that any solid sphere can be split into finitely many subsets which can themselves be reassembled to form two solid spheres, each of the same size as the original [17]. Questions regarding the independence of AC appeared naturally. In 1922, Fraenkel introduced the permutation method to prove the independence of AC from a set theory with atoms [26]. In 1935-1940, Gödel proved that AC is consistent (it does not induce a contradiction) with the axioms of von Neumann/Bernays/Gödel set theory [31]. In 1963, Cohen proved the independence of AC (i.e. the consistency of both AC and its negation) from the standard axioms of ZF set theory, using the so called forcing method which is derived from the Fraenkel’s original permutation method [37]. The original permutation models of Zermelo-Fraenkel set theory with atoms (ZFA) has been recently rediscovered and extended by Gabbay (in a new axiomatic set theoretical framework) and Pitts (in a ZF alternative categorical framework) [29, 44] in order to solve various problems regarding binding, renamings and fresh names in computer science. The alternative set theory introduced by Gabbay and Pitts was extended by Alexandru and Ciobanu [7] in order to describe those algebraic structures that are are defined with respect to the finite support requirement, and by Bojanczyk, Klin and Lasota [23] by considering the so called ‘orbit-finite sets’ that replace ‘finite sets’ in order to solve problems regarding automata, programming languages and Turing machines that operate over infinite alphabets. 4 1 The World of Structures with Finite Supports