On the Finitizability Problem in Algebraic Logic; Recent Results
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The Finitizability Problem in Algebraic Logic, recent results and developments: From neat embeddings to Erdos’ graphs Tarek Sayed Ahmed Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt. October 8, 2018 Abstract . This is an article on the so-called Finitizability Problem in Algebraic Logic. We take a magical tour from the early works of Tarski on relation algebras in the forties all the way to neat embeddings and recent resuts in algebraic logic using Erdos probabilistic graphs. Several deep theorems proved for cylindric algebras are surveyed refined and slightly generalized to other algebraisations of first order logic, like polyadic algebras and diagonal free cylindric algebras. A hitherto unpublished presentation of this problem in a categorial setting is presented. Techniques from stability theory are applied to representation problems in algebraic logic. Philosoph- ical implications are extensively discussed. 1 Algebraic logic starts from certain special logical considerations, abstracts from them, places them in a general algebraic context and via this generaliza- arXiv:1302.1368v1 [math.LO] 6 Feb 2013 tion makes contact with other branches of mathematics (like set theory and topology). It cannot be overemphasized that algebraic logic is more algebra than logic, nor more logic than algebra; in this paper we argue that algebraic logic, particularly the theory of cylindric algebras, has become sufficiently in- teresting and deep to acquire a distinguished status among other subdisciplines of mathematical logic. The principal ideas of the theory of cylindric algebras which is the algebraic setting of first order logic were elaborated by Tarski in cooperation with his students L. H. Chin and F. B. Thompson during the period 1948 - 1952. This was a natural outcome of Tarski’s formalization of the notion of truth in set theory, for indeed the prime examples of cylindric algebras are those algebras whose elements are sets of sequences (i.e., relations) satisfying first 1 2000 Mathematics Subject Classification. Primary 03G15. Key words: algebraic logic, polyadic algebras, amalgamation 1 order formulas. Tarski envisaged that cylindric algebras to first order logic, will be like Boolean algebras to sentential logic. The idea of solving problems in logic by first translating them to algebra, then using the powerful methodology of algebra for solving them, and then translating the solution back to logic, goes back to Leibnitz and Pascal. Pa- pers on the history of Logic (e.g., Anellis - Houser [2], Maddux [78]) allert us to the fact that this method was fruitfully applied in the 19th century with the work of Boole, De Morgan, Peirce, Schr¨oder, etc., on classical logic, see [2]. Employing the similarity between logical equivalence and equality, those pioneers developed logical systems in which metalogical investigations take on a plainly algebraic character. Boole’s work evolved into the modern theory of Boolean algebras, and that of De Morgan, Peirce and Schr¨oder led to but did not end with the theory of relation algebras. From the beginning of the con- temporary era of logic there were two approaches to the subject, one centered on the notion of logical equivalence and the other, reinforced by Hilbert’s work on metamathematics, centered on the notions of assertion and inference. It was not until much later that logicians started to think about connections between these two ways of looking at logic. Tarski [178] gave the precise connection be- tween Boolean algebra and the classical propositional calculus. His approach builds on Lindenbaum’s idea of viewing the set of formulas as an algebra with operations induced by the logical connectives. Logical equivalence is a congru- ence relation on the formula algebra. This is the so-called Lindenbaum-Tarski method. When Tarski applied the this method to the predicate calculus, it led him naturally to the concept of cylindric algebras. Also, we can see that traditionally algebraic logic has focused on the al- gebraic investigation of particular classes of algebras of logic, whether or not they could be connected to some known assertional system by means of the Lindenbaum- Tarski method. However, when such a connection could be es- tablished, there was interest in investigating the relationship between various metalogical properties of the logistic system and the algebraic properties of the associated class of algebras (obtaining what are sometimes called ”bridge theorems”). For example, it was discovered that there is a natural relation between the interpolation theorems of classical, intuitionistic, intermediate propositional calculi, and the amalgamation properties of varieties of Heyting algebras. Similar connections were investigated between interpolation theo- rems in the predicate calculus and amalgamation results in varieties of cylindric and polyadic algebras. Henkin began working with Tarski on the subject of cylindric algebras in the fifties, and a report of their joint research appeared in 1961. By then Monk had also made substantial contributions to the theory. The three planned to write a comprehensive two-volume treatise on the theory of cylindric algebras. The first volume treated cylindric algebras from a general algebraic point of 2 view, while the second volume contained other topics, such as the representa- tion theory, to which Andr´eka and N´emeti contributed a lot, and connections between cylindric algebras and logic. We can find that the theory of cylindric algebras is explicated primarily in three substantial monographs : Henkin, Monk and Tarski [36], [37], and Henkin, Monk, Tarski, Andreka and Nemeti [38]. This covers the development of the subject till the mid eightees of the last century. This paper surveys and refines later developments of the subject. Highlighting the connections with graph theory, model theory, set theory, fi- nite combinatorics, the paper presents topics of broad interest in a way that is accessible to a large audience. The paper is not only purely expository, for it contains, in addition, new ideas and results and also new approaches to old ones. We hope that this paper also provides rapid dissemination of the latest research in the field. A cylindric algebra consists of a Boolean algebra endowed with an addi- tional structure consisting of distinguished elements and operations, satisfying a certain system of equational axioms. The introduction and study of these algebras has its motivation in two parts of mathematics: the deductive systems of first-order logic, and a portion of elementary set theory dealing with spaces of various dimensions, better known as cylindric set algebras. Cylindric set algebras are algebras whose elements are relations of a cer tain pre-assigned arity, endowed with set-theoretic operations that utilize the form of elements of the algebra as sets of sequences. Our notation is in con- formity with the monograph [36], [37]. B(X) denotes the boolean set algebra (℘(X), ∪, ∩, ∼, ∅,X). Let U be a set and α an ordinal. α will be the dimen- α sion of the algebra. For s, t ∈ U write s ≡i t if s(j) = t(j) for all j =6 i. For X ⊆ αU and i, j < α, let α CiX = {s ∈ U : ∃t ∈ X(t ≡i s)} and α Dij = {s ∈ U : si = sj}. α (B( U), Ci, Dij)i,j<α is called the full cylindric set algebra of dimension α with unit (or greatest element) αU. Examples of subalgebras of such set algebras arise naturally from models of first order theories. Indeed if M is a first order structure in a first order language L with α many variables, then one manufactures a cylindric set algebra based on M as follows. Let φM = {s ∈ αM : M |= φ[s]}, (here M |= φ[s] means that s satisfies φ in M), then the set {φM : φ ∈ F mL} is a cylindric set algebra of dimension α. Indeed φM ∩ ψM =(φ ∧ ψ)M, 3 and αM ∼ φM =(¬φ)M, M M Ci(φ )= ∃viφ , and finally M Dij =(xi = xj) . Csα denotes the class of all subalgebras of full set algebras of dimension α. CAα stands for the class of cylindric algebras of dimension α. This is obtained from cylindric set algebras by a process of abstraction and is defned bya finite schema of equations that hold of course in the more concrete set algebra. Definition 0.1. By a cylindric algebra of dimension α, briefly a CAα, we mean an algebra A =(A, +, ·, −, 0, 1, ci, dij)κ,λ<α where (A, +, ·, −, 0, 1) is a boolean algebra such that 0, 1, and dij are distin- guished elements of A (for all j,i < α), − and ci are unary operations on A (for all i<α), + and . are binary operations on A, and such that the following postulates are satisfies for any x, y ∈ A and any i,j,µ <α: (C1) ci0 = 0, (C2) x ≤ cix (i.e., x + cix = cix), (C3) ci(x · ciy)= cix · ciy, (C4) cicjx = cjcix, (C5) dii = 1, (C6) if i =6 j, µ, then djµ = ci(dji · diµ), (C7) if i =6 j, then ci(dij · x) · ci(dij · −x) = 0. A ∈ CAω is locally finite, if the dimension set of every element x ∈ A is finite. The dimension set of x, or ∆x for short, is the set {i ∈ ω : cix =6 x}. Tarski proved that every locally finite ω-dimensional cylindric algebra is rep- resentable, i.e. isomorphic to a subdirect product of set algebra each of di- mension ω. Let Lf ω denote the class of locally finite cylindric algebras. Let RCAω stand for the class of isomorphic copies of subdirect products of set al- gebras each of dimension ω, or briefly, the class of ω dimensional representable cylindric algebras. Then Tarski’s theorem reads Lf ω ⊆ RCAω. This repre- sentation theorem is non-trivial; in fact it is equivalent to G¨odel’s celebrated Completeness Theorem [37, §4.3].