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Journal articles on the topic 'Mathematics Proof theory. Logic, Symbolic and mathematical'

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Published: 4 June 2021
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1

Eklof, Paul C. "Fred Appenzeller. An independence result in quadratic form theory: infinitary combinatorics applied to ε-Hermitian spaces. The journal of symbolic logic, vol. 54 (1989), pp. 689–699. - Otmar Spinas. Linear topologies on sesquilinear spaces of uncountable dimension. Fundamenta mathematicae, vol. 139 (1991), pp. 119–132. - James E. Baumgartner, Matthew Foreman, and Otmar Spinas. The spectrum of the Γ-invariant of a bilinear space. Journal of algebra, vol. 189 (1997), pp. 406–418. - James E. Baumgartner and Otmar Spinas. Independence and consistency proofs in quadratic form theory. The journal of symbolic logic, vol. 56 (1991), pp. 1195–1211. - Otmar Spinas. Iterated forcing in quadratic form theory. Israel journal of mathematics, vol. 79 (1992), pp. 297–315. - Otmar Spinas. Cardinal invariants and quadratic forms. Set theory of the reals, edited by Haim Judah, Israel mathematical conference proceedings, vol. 6, Gelbart Research Institute for Mathematical Sciences, Bar-Ilan University, Ramat-Gan 1993, distributed by the American Mathematical Society, Providence, pp. 563–581. - Saharon Shelah and Otmar Spinas. Gross spaces. Transactions of the American Mathematical Society, vol. 348 (1996), pp. 4257–4277." Bulletin of Symbolic Logic 7, no. 2 (June 2001): 285–86. http://dx.doi.org/10.2307/2687785.

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Buss, Samuel, Ulrich Kohlenbach, and Michael Rathjen. "Mathematical Logic: Proof Theory, Constructive Mathematics." Oberwolfach Reports 8, no. 4 (2011): 2963–3002. http://dx.doi.org/10.4171/owr/2011/52.

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Buss, Samuel, Ulrich Kohlenbach, and Michael Rathjen. "Mathematical Logic: Proof Theory, Constructive Mathematics." Oberwolfach Reports 11, no. 4 (2014): 2933–86. http://dx.doi.org/10.4171/owr/2014/52.

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Buss, Samuel, Rosalie Iemhoff, Ulrich Kohlenbach, and Michael Rathjen. "Mathematical Logic: Proof Theory, Constructive Mathematics." Oberwolfach Reports 14, no. 4 (December 18, 2018): 3121–83. http://dx.doi.org/10.4171/owr/2017/53.

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Buss, Samuel, Rosalie Iemhoff, Ulrich Kohlenbach, and Michael Rathjen. "Mathematical Logic: Proof Theory, Constructive Mathematics." Oberwolfach Reports 17, no. 4 (September 13, 2021): 1693–757. http://dx.doi.org/10.4171/owr/2020/34.

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Arai, Toshiyasu. "Wilfried Buchholz. Notation systems for infinitary derivations. Archive for mathematical logic, vol. 30 no. 5–6 (1991), pp. 277–296. - Wilfried Buchholz. Explaining Gentzen's consistency proof within infinitary proof theory. Computational logic and proof theory, 5th Kurt Gödel colloquium, KGC '97, Vienna, Austria, August 25–29, 1997, Proceedings, edited by Georg Gottlob, Alexander Leitsch, and Daniele Mundici, Lecture notes in computer science, vol. 1289, Springer, Berlin, Heidelberg, New York, etc., 1997, pp. 4–17. - Sergei Tupailo. Finitary reductions for local predicativity, I: recursively regular ordinals. Logic Colloquium '98, Proceedings of the annual European summer meeting of the Association for Symbolic Logic, held in Prague, Czech Republic, August 9–15, 1998, edited by Samuel R. Buss, Petr Háajek, and Pavel Pudlák, Lecture notes in logic, no. 13, Association for Symbolic Logic, Urbana, and A K Peters, Natick, Mass., etc., 2000, pp. 465–499." Bulletin of Symbolic Logic 8, no. 3 (September 2002): 437–39. http://dx.doi.org/10.2178/bsl/1182353905.

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Avigad, Jeremy. "Forcing in Proof Theory." Bulletin of Symbolic Logic 10, no. 3 (September 2004): 305–33. http://dx.doi.org/10.2178/bsl/1102022660.

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AbstractPaul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments.
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Gentilini, Paolo. "Proof theory and mathematical meaning of paraconsistent C-systems." Journal of Applied Logic 9, no. 3 (September 2011): 171–202. http://dx.doi.org/10.1016/j.jal.2011.04.001.

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NEGRI, SARA, and JAN VON PLATO. "Proof systems for lattice theory." Mathematical Structures in Computer Science 14, no. 4 (August 2004): 507–26. http://dx.doi.org/10.1017/s0960129504004244.

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A formulation of lattice theory as a system of rules added to sequent calculus is given. The analysis of proofs for the contraction-free calculus of classical predicate logic known as G3c extends to derivations with the mathematical rules of lattice theory. It is shown that minimal derivations of quantifier-free sequents enjoy a subterm property: all terms in such derivations are terms in the endsequent.An alternative formulation of lattice theory as a system of rules in natural deduction style is given, both with explicit meet and join constructions and as a relational theory with existence axioms. A subterm property for the latter extends the standard decidable classes of quantificational formulas of pure predicate calculus to lattice theory.
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RABE, FLORIAN. "A logical framework combining model and proof theory." Mathematical Structures in Computer Science 23, no. 5 (March 1, 2013): 945–1001. http://dx.doi.org/10.1017/s0960129512000424.

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Mathematical logic and computer science have driven the design of a growing number of logics and related formalisms such as set theories and type theories. In response to this population explosion, logical frameworks have been developed as formal meta-languages in which to represent, structure, relate and reason about logics.Research on logical frameworks has diverged into separate communities, often with conflicting backgrounds and philosophies. In particular, two of the most important logical frameworks are the framework of institutions, from the area of model theory based on category theory, and the Edinburgh Logical Framework LF, from the area of proof theory based on dependent type theory. Even though their ultimate motivations overlap – for example in applications to software verification – they have fundamentally different perspectives on logic.In the current paper, we design a logical framework that integrates the frameworks of institutions and LF in a way that combines their complementary advantages while retaining the elegance of each of them. In particular, our framework takes a balanced approach between model theory and proof theory, and permits the representation of logics in a way that comprises all major ingredients of a logic: syntax, models, satisfaction, judgments and proofs. This provides a theoretical basis for the systematic study of logics in a comprehensive logical framework. Our framework has been applied to obtain a large library of structured and machine-verified encodings of logics and logic translations.
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Harnik, Victor, and Michael Makkai. "Lambek's categorical proof theory and Läuchli's abstract realizability." Journal of Symbolic Logic 57, no. 1 (March 1992): 200–230. http://dx.doi.org/10.2307/2275186.

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In this paper we give an introduction to categorical proof theory, and reinterpret, with improvements, Läuchli's work on abstract realizability restricted to propositional logic (but see [M1] for predicate logic). Partly to make some points of a foundational nature, we have included a substantial amount of background material. As a result, the paper is (we hope) readable with a knowledge of just the rudiments of category theory, the notions of category, functor, natural transformation, and the like. We start with an extended introduction giving the background, and stating what we do with a minimum of technicalities.In three publications [L1, 2, 3] published in the years 1968, 1969 and 1972, J. Lambek gave a categorical formulation of the notion of formal proof in deductive systems in certain propositional calculi. The theory is also described in the recent book [LS]. See also [Sz].The basic motivation behind Lambek's theory was to place proof theory in the framework of modern abstract mathematics. The spirit of the latter, at least for the purposes of the present discussion, is to organize mathematical objects into mathematical structures. The specific kind of structure we will be concerned with is category.In Lambek's theory, one starts with an arbitrary theory in any one of several propositional calculi. One has the (formal) proofs (deductions) in the given theory of entailments A ⇒ B, with A and B arbitrary formulas. One introduces an equivalence relation on proofs under which, in particular, equivalent proofs are proofs of the same entailment; equivalence of proofs is intended to capture the idea of the proofs being only inessentially different. One forms a category whose objects are the formulas of the underlying language of the theory, and whose arrows from A to B, with the latter arbitrary formulas, are the equivalence classes of formal proofs of A ⇒ B.
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Nyikos, Peter J. "Andreas Blass and Saharon Shelah. Ultrafilters with small generating sets. Israel journal of mathematics, vol. 65 (1989), pp. 259–271. - Andreas Blass and Saharon Shelah. There may be simple - and -points and the Rudin–Keisler ordering may be downward directed. Annals of pure and applied logic, vol. 33 (1987), pp. 213–243. - Andreas Blass. Near coherence of filters. II: Applications to operator ideals, the Stone–Čech remainder of a half-line, order ideals of sequences, and the slenderness of groups. Transactions of the American Mathematical Society, vol. 300 (1987), pp. 557–581. - Andreas Blass and Saharon Shelah. Near coherence of filters III: a simplified consistency proof. Notre Dame journal of formal logic, vol. 30 (1989), pp. 530–538. - Andreas Blass and Claude Laflamme. Consistency results about filters and the number of inequivalent growth types. The journal of symbolic logic, vol. 54 (1989), pp. 50–56. - Andreas Blass. Applications of superperfect forcing and its relatives. Set theory and its applications. Proceedings of a conference held at York University, Ontario, Canada, Aug. 10–21, 1987, edited by J. Steprāns and S. Watson, Lecture notes in mathematics, vol. 1401, Springer-Verlag, Berlin etc. 1989, pp. 18–40. - Andreas Blass and Saharon Shelah. Ultrafilters with small generating sets. Israel journal of mathematics, vol. 65 (1989), pp. 259–271." Journal of Symbolic Logic 57, no. 2 (June 1992): 763–66. http://dx.doi.org/10.2307/2275316.

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13

Turchin, Valentin F. "A constructive interpretation of the full set theory." Journal of Symbolic Logic 52, no. 1 (March 1987): 172–201. http://dx.doi.org/10.2307/2273872.

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The interpretation of the ZF set theory reported in this paper is, actually, part of a wider effort, namely, a new approach to the foundation of mathematics, which is referred to as The Cybernetic Foundation. A detailed exposition of the Cybernetic Foundation will be published elsewhere. Our approach leads to a full acceptance of the formalism of the classical set theory, but interprets it using only the idea of potential, but not actual (completed) infinity, and dealing only with finite objects that can actually be constructed. Thus we have a finitist proof of the consistency of ZF. This becomes possible because we set forth a metatheory of mathematics which goes beyond the classical logic and set theory and, of course, cannot be formalized in ZF, yet yields proofs which are as convincing—at least, from the author's viewpoint—as any mathematical proof can be.Our metatheory is based on the following two ideas. Firstly, we define the semantics of the mathematical language using the cybernetical concept of knowledge. According to this concept, to say that a cybernetic system (a human being, in particular) has some knowledge is to say that it has some models of reality. In the Cybernetic Foundation we consider mathematics as the art of constructing linguistic models of reality.
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BALDWIN, JOHN T. "FORMALIZATION, PRIMITIVE CONCEPTS, AND PURITY." Review of Symbolic Logic 6, no. 1 (September 19, 2012): 87–128. http://dx.doi.org/10.1017/s1755020312000263.

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AbstractWe emphasize the role of the choice of vocabulary in formalization of a mathematical area and remark that this is a particular preoccupation of logicians. We use this framework to discuss Kennedy’s notion of ‘formalism freeness’ in the context of various schools in model theory. Then we clarify some of the mathematical issues in recent discussions of purity in the proof of the Desargues proposition. We note that the conclusion of ‘spatial content’ from the Desargues proposition involves arguments which are algebraic and even metamathematical. Hilbert showed that the Desargues proposition implies the coordinatizing ring is associative, which in turn implies the existence of a three-dimensional geometry in which the given plane can be embedded. With W. Howard we give a new proof, removing Hilbert’s ‘detour’ through algebra, of the ‘geometric’ embedding theorem.Finally, our investigation of purity leads to the conclusion that even the introduction of explicit definitions in a proof can violate purity. We argue that although both involve explicit definition, our proof of the embedding theorem is pure while Hilbert’s is not. Thus the determination of whether an argument is pure turns on the content of the particular proof. Moreover, formalizing the situation does not provide a tool for characterizing purity.
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Pohlers, Wolfram. "Pure Proof Theory Aims, Methods and Results: Extended Version of Talks Given at Oberwolfach and Haifa." Bulletin of Symbolic Logic 2, no. 2 (June 1996): 159–88. http://dx.doi.org/10.2307/421108.

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Apologies. The purpose of the following talk is to give an overview of the present state of aims, methods and results in Pure Proof Theory. Shortage of time forces me to concentrate on my very personal views. This entails that I will emphasize the work which I know best, i.e., work that has been done in the triangle Stanford, Munich and Münster. I am of course well aware that there are as important results coming from outside this triangle and I apologize for not displaying these results as well.Moreover the audience should be aware that in some points I have to oversimplify matters. Those who complain about that are invited to consult the original papers.1.1. General. Proof theory startedwithHilbert's Programme which aimed at a finitistic consistency proof for mathematics.By Gödel's Theorems, however, we know that we can neither formalize all mathematics nor even prove the consistency of formalized fragments by finitistic means. Inspite of this fact I want to give some reasons why I consider proof theory in the style of Gentzen's work still as an important and exciting field of Mathematical Logic. I will not go into applications of Gentzen's cut-elimination technique to computer science problems—this may be considered as applied proof theory—but want to concentrate on metamathematical problems and results. In this sense I am talking about Pure Proof Theory.Mathematicians are interested in structures. There is only one way to find the theorems of a structure. Start with an axiom system for the structure and deduce the theorems logically. These axiom systems are the objects of proof-theoretical research. Studying axiom systems there is a series of more or less obvious questions.
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Calude, Cristian S., and Elena Calude. "The complexity of the four colour theorem." LMS Journal of Computation and Mathematics 13 (August 27, 2010): 414–25. http://dx.doi.org/10.1112/s1461157009000461.

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AbstractThe four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a non-trivial computer verification. Recently, a formal proof of the theorem was obtained with the equational logic program Coq [G. Gonthier, ‘Formal proof–the four color theorem’,Notices of Amer. Math. Soc.55 (2008) no. 11, 1382–1393]. In this paper we describe an implementation of the computational method introduced by C. S. Calude and co-workers [Evaluating the complexity of mathematical problems. Part 1’,Complex Systems18 (2009) 267–285; A new measure of the difficulty of problems’,J. Mult. Valued Logic Soft Comput.12 (2006) 285–307] to evaluate the complexity of the four colour theorem. Our method uses a Diophantine equational representation of the theorem. We show that the four colour theorem is in the complexity class ℭU,4. For comparison, the Riemann hypothesis is in class ℭU,3while Fermat’s last theorem is in class ℭU,1.
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Drápal, Aleš. "Richard Laver. The left distributive law and the freeness of an algebra of elementary embeddings. Advances in mathematics, vol. 91 (1992), pp. 209–231. - Richard Laver. A division algorithm for the free left distributive algebra. Logic Colloquium '90, ASL summer meeting in Helsinki, edited by J. Oikkonen and J. Väänänen, Lecture notes in logic, no. 2, Springer-Verlag, Berlin, Heidelberg, New York, etc., 1993, pp. 155–162. - Richard Laver. On the algebra of elementary embeddings of a rank into itself. Advances in mathematics, vol. 110 (1995), pp. 334–346. - Richard Laver. Braid group actions on left distributive structures, and well orderings in the braid groups. Journal of pure and applied algebra, vol. 108 (1996), pp. 81–98. - Patrick Dehornoy. An alternative proof of Laver's results on the algebra generated by an elementary embedding. Set theory of the continuum, edited by H. Judah, W. Just, and H. Woodin, Mathematics Sciences Research Institute publications, vol. 26, Springer-Verlag, New York, Berlin, Heidelberg, etc., 1992, pp. 27–33. - Patrick Dehornoy. Braid groups and left distributive operations. Transactions of the American Mathematical Society, vol. 345 (1994), pp. 115–150. - Patrick Dehornoy. A normal form for the free left distributive law. International journal of algebra and computation, vol. 4 (1994), pp. 499–528. - Patrick Dehornoy. From large cardinals to braids via distributive algebra. Journal of knot theory and its ramifications, vol. 4 (1995), pp. 33–79. - J. R. Steel. The well-foundedness of the Mitchell order. The journal of symbolic logic, vol. 58 (1993), pp. 931–940. - Randall Dougherty. Critical points in an algebra of elementary embeddings. Annals of pure and applied logic, vol. 65 (1993), pp. 211–241. - Randall Dougherty. Critical points in an algebra of elementary embeddings, II. Logic: from foundations to applications, European logic colloquium, edited by Wilfrid Hodges, Martin Hyland, Charles Steinhorn, and John Truss, Clarendon Press, Oxford University Press, Oxford, New York, etc., 1996, pp. 103–136. - Randall Dougherty and Thomas Jech. Finite left-distributive algebras and embedding algebras. Advances in mathematics, vol. 130 (1997), pp. 201–241." Bulletin of Symbolic Logic 8, no. 4 (December 2002): 555–60. http://dx.doi.org/10.2178/bsl/1182353941.

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18

Shalack, V. I. "On First-order Theories Which Can Be Represented by Definitions." Logical Investigations 22, no. 1 (March 3, 2016): 125–35. http://dx.doi.org/10.21146/2074-1472-2016-22-1-125-135.

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In the paper we consider the classical logicism program restricted to first-order logic.The main result of this paper is the proof of the theorem, which contains the necessary and sufficient conditions for a mathematical theory to be reducible to logic. Those and only those theories, which don’t impose restrictions on the size of their domains, can be reduced to pure logic. Among such theories we can mention the elementary theory of groups, the theory of combinators (combinatory logic), the elementary theory of topoi and many others. It is interesting to note that the initial formulation of the problem of reduction of mathematics to logic is principally insoluble. As we know all theorems of logic are true in the models with any number of elements. At the same time, many mathematical theories impose restrictions on size of their models. For example, all models of arithmetic have an infinite number of elements. If arithmetic was reducible to logic, it would had finite models, including an one-element model. But this is impossible in view of the axiom $0 \neq x'$.
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Zeman, Martin. "Ernest Schimmerling. Covering properties of core models. Sets and proofs. (Leeds, 1997), London Mathematical Society Lecture Note Series 258. Cambridge University Press, Cambridge, 1999, pp. 281–299. - Peter Koepke. An introduction to extenders and core models for extender sequences. Logic Colloquium '87 (Granada, 1987), Studies in Logic and the Foundations of Mathematics 129. North-Holland, Amsterdam, 1989, pp. 137–182. - William J. Mitchell. The core model up to a Woodin cardinal. Logic, methodology and philosophy of science, IX (Uppsala, 1991), Studies in Logic and the Foundations of Mathematics 134, North-Holland, Amsterdam, 1994, pp. 157–175. - Benedikt Löwe and John R. Steel. An introduction to core model theory. Sets and proofs (Leeds, 1997), London Mathematical Society Lecture Note Series 258, Cambridge University Press, Cambridge, 1999, pp. 103–157. - John R. Steel. Inner models with many Woodin cardinals. Annals of Pure and Applied Logic, vol. 65 no. 2 (1993), pp. 185–209. - Ernest Schimmerling. Combinatorial principles in the core model for one Woodin cardinal. Annals of Pure and Applied Logic, vol. 74 no. 2 (1995), pp. 153–201. - Philip D. Welch. Some remarks on the maximality of inner models. Logic Colloquium '98 (Prague, 1998), Lecture Notes in Logic 13, Association of Symbolic Logic, Urbana, Illinois, 2000, pp. 516–540." Bulletin of Symbolic Logic 10, no. 4 (December 2004): 583–88. http://dx.doi.org/10.1017/s1079898600003681.

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Kanamori, Akihiro. "Gödel and Set Theory." Bulletin of Symbolic Logic 13, no. 2 (June 2007): 153–88. http://dx.doi.org/10.2178/bsl/1185803804.

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Kurt Gödel (1906–1978) with his work on the constructible universeLestablished the relative consistency of the Axiom of Choice (AC) and the Continuum Hypothesis (CH). More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic constructions and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel's work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges are the roots and anticipations in work of Russell and Hilbert, and most prominently the sustained motif of truth as formalizable in the “next higher system”. We especially work at bringing out how transforming Gödel's work was for set theory. It is difficult now to see what conceptual and technical distance Gödel had to cover and how dramatic his re-orientation of set theory was.
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Roman’kov, V. A. "Algorithmic theory of solvable groups." Prikladnaya Diskretnaya Matematika, no. 52 (2021): 16–64. http://dx.doi.org/10.17223/20710410/52/2.

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The purpose of this survey is to give some picture of what is known about algorithmic and decision problems in the theory of solvable groups. We will provide a number of references to various results, which are presented without proof. Naturally, the choice of the material reported on reflects the author’s interests and many worthy contributions to the field will unfortunately go without mentioning. In addition to achievements in solving classical algorithmic problems, the survey presents results on other issues. Attention is paid to various aspects of modern theory related to the complexity of algorithms, their practical implementation, random choice, asymptotic properties. Results are given on various issues related to mathematical logic and model theory. In particular, a special section of the survey is devoted to elementary and universal theories of solvable groups. Special attention is paid to algorithmic questions regarding rational subsets of groups. Results on algorithmic problems related to homomorphisms, automorphisms, and endomorphisms of groups are presented in sufficient detail.
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AWODEY, STEVE, and MICHAEL A. WARREN. "Homotopy theoretic models of identity types." Mathematical Proceedings of the Cambridge Philosophical Society 146, no. 1 (January 2009): 45–55. http://dx.doi.org/10.1017/s0305004108001783.

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Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and—as, e.g., the work of Voevodsky on the homotopy theory of schemes [15] or the work of Joyal [11,12] and Lurie [13] on quasicategories seem to indicate—it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, inspired by the groupoid model of (intensional) Martin–Löf type theory [14] due to Hofmann and Streicher [9]. In particular, we show that a form of Martin–Löf type theory can be soundly modelled in any model category. This result indicates moreover that any model category has an associated “internal language” which is itself a form of Martin-Löf type theory. This suggests applications both to type theory and to homotopy theory. Because Martin–Löf type theory is, in one form or another, the theoretical basis for many of the computer proof assistants currently in use, such asCoqandAgda(cf. [3] and [5]), this promise of applications is of a practical, as well as theoretical, nature.
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Ovsyak, V. K., O. V. Ovsyak, and J. V. Petruszka. "ORDER AND ORDERING IN DISCRETE MATHEMATICS AND INFORMATICS." Ukrainian Journal of Information Technology 3, no. 1 (2021): 37–43. http://dx.doi.org/10.23939/ujit2021.03.037.

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The available means of ordering and sorting in some important sections of discrete mathematics and computer science are studied, namely: in the set theory, classical mathematical logic, proof theory, graph theory, POST method, system of algorithmic algebras, algorithmic languages of object-oriented and assembly programming. The Cartesian product of sets, ordered pairs and ordered n-s, the description by means of set theory of an ordered pair, which are performed by Wiener, Hausdorff and Kuratowski, are presented. The requirements as for the relations that order sets are described. The importance of ordering in classical mathematical logic and proof theory is illustrated by the examples of calculations of the truth values of logical formulas and formal derivation of a formula on the basis of inference rules and substitution rules. Ordering in graph theory is shown by the example of a block diagram of the Euclidean algorithm, designed to find the greatest common divisor of two natural numbers. The ordering and sorting of both the instructions formed by two, three and four ordered fields and the existing ordering of instructions in the program of Post method are described. It is shown that the program is formed by the numbered instructions with unique instruction numbers and the presence of the single instruction with number 1. The means of the system of algorithmic algebras, which are used to perform the ordering and sorting in the algorithm theory, are illustrated. The operations of the system of algorithmic algebras are presented, which include Boolean algebra operations generalized to the three-digit alphabet and operator operations of operator algebra. The properties of the composition operation are described, which is intended to describe the orderings of the operators of the operator algebra in the system of algorithmic algebras. The orderings executed by means of algorithmic programming languages are demonstrated by the hypothetical application of the modern object-oriented programming language C#. The program must contain only one method Main () from which the program execution begins. The ARM microprocessor assembly program must have only one ENTRY directive from which the program execution begins.
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Harnik, Victor. "Stability theory and set existence axioms." Journal of Symbolic Logic 50, no. 1 (March 1985): 123–37. http://dx.doi.org/10.2307/2273795.

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A series of investigations started by H. Friedman ([5], [6]) and pursued by him, by S. Simpson, and by others (see [7], [16] and the references there) had the objective of determining what are precisely the set existence axioms needed for proving theorems of “ordinary” mathematics. The study centered around -CA0 (for “-Comprehension Axiom”), a fragment of second order arithmetic in which the main body of “ordinary” mathematics can be comfortably developed (of course, after a suitable encoding of the various concepts into numbers and sets of numbers). The main question is always this: given a theorem τ provable in -CA0, do we need all of -CA0, or, maybe, is τ provable in a weaker subsystem? A surprisingly simple pattern emerged: there seem to be just five important systems, denoted (in order of increasing strength) RCA0, WKL0, ACA0, ATR0 and -CA0, such that in most cases, whenever an ordinary mathematical theorem τ is provable in -CA0, it is either provable in RCA0 or it is equivalent to one, call it S, of the other four systems, the proof of the equivalence being done in one of the systems which is weaker than S (most typically, RCA0 or ACA0). This very interesting phenomenon—or “theme” as it was called by Friedman—has been verified for many instances in the realm of analysis and algebra (cf. the references above and the forthcoming book [17]). It is the purpose of this paper to do the same for a particular branch of model theory, namely stability theory.
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HUET, GÉRARD. "Special issue on ‘Logical frameworks and metalanguages’." Journal of Functional Programming 13, no. 2 (March 2003): 257–60. http://dx.doi.org/10.1017/s0956796802004549.

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There is both a great unity and a great diversity in presentations of logic. The diversity is staggering indeed – propositional logic, first-order logic, higher-order logic belong to one classification; linear logic, intuitionistic logic, classical logic, modal and temporal logics belong to another one. Logical deduction may be presented as a Hilbert style of combinators, as a natural deduction system, as sequent calculus, as proof nets of one variety or other, etc. Logic, originally a field of philosophy, turned into algebra with Boole, and more generally into meta-mathematics with Frege and Heyting. Professional logicians such as Gödel and later Tarski studied mathematical models, consistency and completeness, computability and complexity issues, set theory and foundations, etc. Logic became a very technical area of mathematical research in the last half century, with fine-grained analysis of expressiveness of subtheories of arithmetic or set theory, detailed analysis of well-foundedness through ordinal notations, logical complexity, etc. Meanwhile, computer modelling developed a need for concrete uses of logic, first for the design of computer circuits, then more widely for increasing the reliability of sofware through the use of formal specifications and proofs of correctness of computer programs. This gave rise to more exotic logics, such as dynamic logic, Hoare-style logic of axiomatic semantics, logics of partial values (such as Scott's denotational semantics and Plotkin's domain theory) or of partial terms (such as Feferman's free logic), etc. The first actual attempts at mechanisation of logical reasoning through the resolution principle (automated theorem proving) had been disappointing, but their shortcomings gave rise to a considerable body of research, developing detailed knowledge about equational reasoning through canonical simplification (rewriting theory) and proofs by induction (following Boyer and Moore successful integration of primitive recursive arithmetic within the LISP programming language). The special case of Horn clauses gave rise to a new paradigm of non-deterministic programming, called Logic Programming, developing later into Constraint Programming, blurring further the scope of logic. In order to study knowledge acquisition, researchers in artificial intelligence and computational linguistics studied exotic versions of modal logics such as Montague intentional logic, epistemic logic, dynamic logic or hybrid logic. Some others tried to capture common sense, and modeled the revision of beliefs with so-called non-monotonic logics. For the careful crafstmen of mathematical logic, this was the final outrage, and Girard gave his anathema to such “montres à moutardes”.
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Sieg, Wilfried. "Hilbert's Programs: 1917–1922." Bulletin of Symbolic Logic 5, no. 1 (March 1999): 1–44. http://dx.doi.org/10.2307/421139.

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AbstractHilbert's finitist program was not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration with Bernays during the period from 1917 to 1922. These notes reveal a dialectic progression from a critical logicism through a radical constructivism toward finitism; the progression has to be seen against the background of the stunning presentation of mathematical logic in the lectures given during the winter term 1917/18. In this paper, I sketch the connection of Hilbert's considerations to issues in the foundations of mathematics during the second half of the 19th century, describe the work that laid the basis of modern mathematical logic, and analyze the first steps in the new subject of proof theory. A revision of the standard view of Hilbert's and Bernays's contributions to the foundational discussion in our century has long been overdue. It is almost scandalous that their carefully worked out notes have not been used yet to understand more accurately the evolution of modern logic in general and of Hilbert's Program in particular. One conclusion will be obvious: the dogmatic formalist Hilbert is a figment of historical (de)construction! Indeed, the study and analysis of these lectures reveal a depth of mathematical-logical achievement and of philosophical reflection that is remarkable. In the course of my presentation many questions are raised and many more can be explored; thus, I hope this paper will stimulate interest for new historical and systematic work.
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Qudrat-I Elahi, Khandakar. "A difficulty in Arrow’s impossibility theorem." International Journal of Social Economics 44, no. 12 (December 4, 2017): 1609–21. http://dx.doi.org/10.1108/ijse-02-2016-0065.

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Purpose The purpose of this paper is twofold. First, it evaluates the impossibility proposition, called the “Arrow impossibility theorem” (AIT), which is widely attributed to Arrow’s social choice theory. This theorem denies the possibility of arriving at any collective majority resolution in any group voting system if the social choice function must satisfy “certain natural conditions”. Second, it intends to show the reasons behind the proliferation of this impossibility impression. Design/methodology/approach Theoretical and philosophical. Findings Arrow’s mathematical model does not seem to suggest or support his impossibility thesis. He has considered only one voting outcome, well known by the phrase “the Condorcet paradox”. However, other voting results are equally likely from his model, which might suggest unambiguous majority choice. This logical dilemma has been created by Arrow’s excessive dependence on the language of mathematics and symbolic logic. Research limitations/implications The languages of mathematics and symbolic logic – numbers, letters and signs – have definite advantages in scientific argumentation and reasoning. These numbers and letters being invented however do not have any behavioural characteristics, which suggests that conclusions drawn from the model merely reflect the author’s opinions. The AIT is a good example of this logical dilemma. Social implications The modern social choice theory, which is founded on the AIT, seems to be an academic assault to the system of democratic governance that is dominating current global village. By highlighting weaknesses in the AIT, this paper attempts to discredit this intellectual omission. Originality/value The paper offers a counter example to show that the impossibility of social choice is not necessarily implied by the Arrow’s model. Second, it uses Locke’s theory of human understanding to explain why the concerned social scientists are missing this point. This approach is probably entirely novel in this area of research.
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Wang, Xiao Gang. "Significance of Mathematization of Philosophical Problems from the Angle of Broadspectrum Philosophy." Advanced Materials Research 433-440 (January 2012): 6315–18. http://dx.doi.org/10.4028/www.scientific.net/amr.433-440.6315.

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Whether philosophy can realize mathematization has long been controversial. As the mathematics develops a nonquantative branch- structural mathematics, however, mathematization of philosophy has a turnaround. Broadspectrum philosophy which makes use of structural mathematics has established a generally applicable as well as precise mathematical model for many philosophical problems, giving a positive answer to whether the philosophy can be mathematized. Mathematizaiton of philosophy allows more accurate and clear distinction of people’s expression in meaning, gives ideas the visible characteristics, makes philosophy an analyzable discipline, and realizes routinization of philosophical methods. Hegel was well versed in mathematics but opposed “Extreme Mathematic Attitude”, since he thought recognizing all the objects from the mathematic standpoint of “Quantity or Quantitative Relationship” would ignore the qualitative difference among the objects.［1］P239 Hegel’s opinion was based on the traditional mathematic which takes the Quantitative Relationship as the foundation. Holding the same evidence as Hegel's, most philosophers nowadays still suspect that the philosophy can be mathematized. When the modern mathematics has developed a new nonquantative branch, the Structural Mathematics, the philosophy mathematization, however, meets a turning point. Opposed to Quantitative Mathematics, the Structural Mathematics focuses on research of mathematic relationship and structure on the basis of abstract set theory. Since the structural mathematics doesn't rely on quantity and quantitative relationship, it can be combined in research of philosophy which usually doesn’t possess quantitative characteristics. Establishment of Broadspectrum Philosophy is a successful attempt. With full application of set theory, symbolic logic, modern algebra, transformation group theory and graph theory, Broadspectrum Philosophy constructs a generally applicable as well as precise mathematical mode for many philosophical problems, bringing a fundamental change to the philosophy. This paper attempts to make some preliminary analysis on the significance of establishment of Broadspectrum Philosophy.
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Yasuhara, M. "Peter B. Andrews. An introduction to mathematical logic and type theory: to truth through proof. Computer science and applied mathematics. Academic Press, Orlando etc. 1986, xv + 304 pp." Journal of Symbolic Logic 53, no. 1 (March 1988): 312–14. http://dx.doi.org/10.1017/s0022481200029194.

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Gabbay, Murdoch J. "Foundations of Nominal Techniques: Logic and Semantics of Variables in Abstract Syntax." Bulletin of Symbolic Logic 17, no. 2 (June 2011): 161–229. http://dx.doi.org/10.2178/bsl/1305810911.

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AbstractWe are used to the idea that computers operate on numbers, yet another kind of data is equally important: the syntax of formal languages, with variables, binding, and alpha-equivalence. The original application of nominal techniques, and the one with greatest prominence in this paper, is to reasoning on formal syntax with variables and binding.Variables can be modelled in many ways: for instance as numbers (since we usually take countably many of them); as links (since they may ‘point’ to a binding site in the term, where they are bound); or as functions (since they often, though not always, represent ‘an unknown’).None of these models is perfect. In every case for the models above, problems arise when trying to use them as a basis for a fully formal mechanical treatment of formal language. The problems are practical—but their underlying cause may be mathematical.The issue is not whether formal syntax exists, since clearly it does, so much as what kind of mathematical structure it is. To illustrate this point by a parody, logical derivations can be modelled using a Gödel encoding (i.e., injected into the natural numbers). It would be false to conclude from this that proof-theory is a branch of number theory and can be understood in terms of, say, Peano's axioms. Similarly, as it turns out, it is false to conclude from the fact that variables can be encoded e.g., as numbers, that the theory of syntax-with-binding can be understood in terms of the theory of syntax-without-binding, plus the theory of numbers (or, taking this to a logical extreme, purely in terms of the theory of numbers). It cannot; something else is going on. What that something else is, has not yet been fully understood.In nominal techniques, variables are an instance of names, and names are data. We model names using urelemente with properties that, pleasingly enough, turn out to have been investigated by Fraenkel and Mostowski in the first half of the 20th century for a completely different purpose than modelling formal language. What makes this model really interesting is that it gives names distinctive properties which can be related to useful logic and programming principles for formal syntax.Since the initial publications, advances in the mathematics and presentation have been introduced piecemeal in the literature. This paper provides in a single accessible document an updated development of the foundations of nominal techniques. This gives the reader easy access to updated results and new proofs which they would otherwise have to search across two or more papers to find, and full proofs that in other publications may have been elided. We also include some new material not appearing elsewhere.
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Kierstead, Henry A. "G. Metakides and A. Nerode. Recursion theory and algebra. Algebra and logic, Papers from the 1974 Summer Research Institute of the Australian Mathematical Society, Monash University, Australia, edited by J. N. Crossley, Lecture notes in mathematics, vol. 450, Springer-Verlag, Berlin, Heidelberg, and New York, 1975, pp. 209–219. - Iraj Kalantari and Allen Retzlaff. Maximal vector spaces under automorphisms of the lattice of recursively enumerable vector spaces. The journal of symbolic logic, vol. 42 no. 4 (for 1977, pub. 1978), pp. 481–491. - Iraj Kalantari. Major subspaces of recursively enumerable vector spaces. The journal of symbolic logic, vol. 43 (1978), pp. 293–303. - J. Remmel. A r-maximal vector space not contained in any maximal vector space. The journal of symbolic logic, vol. 43 (1978), pp. 430–441. - Allen Retzlaff. Simple and hyperhypersimple vector spaces. The journal of symbolic logic, vol. 43 (1978), pp. 260–269. - J. B. Remmel. Maximal and cohesive vector spaces. The journal of symbolic logic, vol. 42 no. 3 (for 1977, pub. 1978), pp. 400–418. - J. Remmel. On r.e. and co-r.e. vector spaces with nonextendible bases. The journal of symbolic logic, vol. 45 (1980), pp. 20–34. - M. Lerman and J. B. Remmel. The universal splitting property: I. Logic Colloquim '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1982, pp. 181–207. - J. B. Remmel. Recursively enumerable Boolean algebras. Annals of mathematical logic, vol. 15 (1978), pp. 75–107. - J. B. Remmel. r-Maximal Boolean algebras. The journal of symbolic logic, vol. 44 (1979), pp. 533–548. - J. B. Remmel. Recursion theory on algebraic structures with independent sets. Annals of mathematical logic, vol. 18 (1980), pp. 153–191. - G. Metakides and J. B. Remmel. Recursion theory on orderings. I. A model theoretic setting. The journal of symbolic logic, vol. 44 (1979), pp. 383–402. - J. B. Remmel. Recursion theory on orderings. II. The journal of symbolic logic, vol. 45 (1980), pp. 317–333." Journal of Symbolic Logic 51, no. 1 (March 1986): 229–32. http://dx.doi.org/10.2307/2273960.

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Lerman, M. "Carl G. JockuschJr., and David B. Posner. Double jumps of minimal degrees. The journal of symbolic logic, vol. 43 no. 4 (for 1978, pub. 1979), pp. 715–724. - Carl G. JockuschJr., and David B. Posner. Automorphism bases for degrees of unsotvability. Israel journal of mathematics, vol. 40 (1981), pp. 150–164. - Richard L. Epstein. Initial segments of degrees below 0′. Memoirs of the American Mathematical Society, no. 241. American Mathematical Society, Providence1981, vi + 102 pp. - Richard A. Shore. The theory of the degrees below 0′. The journal of the London Mathematical Society, ser. 2 vol. 24 (1981), pp. 1–14." Journal of Symbolic Logic 50, no. 2 (June 1985): 550–52. http://dx.doi.org/10.2307/2274245.

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Kossak, Roman. "James H. Schmerl. Peano models with many generic classes. Pacific Journal of Mathematics, vol. 43 (1973), pp. 523–536. - James H. Schmerl. Correction to: “Peano models with many generic classes”. Pacific Journal of Mathematics, vol. 92 (1981), no. 1, pp. 195–198. - James H. Schmerl. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80 (Proceedings, Seminars, and Conferences in Mathematical Logic, University of Connecticut, Storrs, Connecticut, 1979/80). edited by M. Lerman, J. H. Schmerl, and R. I. Soare, Lecture Notes in Mathematics, vol. 859. Springer, Berlin, pp. 268–282. - James H. Schmerl. Recursively saturatedmodels generated by indiscernibles. Notre Dane Journal of Formal Logic, vol. 26 (1985), no. 1, pp. 99–105. - James H. Schmerl. Large resplendent models generated by indiscernibles. The Journal of Symbolic Logic, vol. 54 (1989), no. 4, pp. 1382–1388. - James H. Schmerl. Automorphism groups of models of Peano arithmetic. The Journal of Symbolic Logic, vol. 67 (2002), no. 4, pp. 1249–1264. - James H. Schmerl. Diversity in substructures. Nonstandard models of arithmetic and set theory. edited by A. Enayat and R. Kossak, Contemporary Mathematics, vol. 361, American Mathematical Societey (2004), pp. 45–161. - James H. Schmerl. Generic automorphisms and graph coloring. Discrete Mathematics, vol. 291 (2005), no. 1–3, pp. 235–242. - James H. Schmerl. Nondiversity in substructures. The Journal of Symbolic Logic, vol. 73 (2008), no. 1, pp. 193–211." Bulletin of Symbolic Logic 15, no. 2 (June 2009): 222–27. http://dx.doi.org/10.1017/s1079898600001359.

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Lubarsky, Robert. "Patrick Farrington. Hinges and automorphisms of the degrees of non-constructibility. The journal of the London Mathematical Society, ser. 2 vol. 28 (1983), pp. 193–202. - Petr Hájek. Some results on degrees of constructibility. Higher set theory, Proceedings, Oberwolfach, Germany, April 13–23, 1977, edited by G. H. Müller and D. S. Scott, Lecture notes in mathematics, vol. 669, Springer-Verlag, Berlin, Heidelberg, and New York, 1978, pp. 55–71. - Zofia Adamowicz. On finite lattices of degrees of constructibility of reals. The journal of symbolic logic, vol. 41 (1976), pp. 313–322. - Zofia Adamowicz. Constructive semi-lattices of degrees of constructibility. Set theory and hierarchy theory V, Bierutowice, Poland 1976, edited by A. Lachlan, M. Srebrny, and A. Zarach, Lecture notes in mathematics, vol. 619, Springer-Verlag, Berlin, Heidelberg, and New York, 1977, pp. 1–43." Journal of Symbolic Logic 54, no. 3 (September 1989): 1109–11. http://dx.doi.org/10.2307/2274781.

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Jockusch, Carl. "Richard A. Shore. Determining automorphisms of the recursively enumerable sets. Proceedings of the American Mathematical Society, vol. 65 (1977), pp. 318– 325. - Richard A. Shore. The homogeneity conjecture. Proceedings of the National Academy of Sciences of the United States of America, vol. 76 (1979), pp. 4218– 4219. - Richard A. Shore. On homogeneity and definability in the first-order theory of the Turing degrees. The journal of symbolic logic, vol. 47 (1982), pp. 8– 16. - Richard A. Shore. The arithmetic and Turing degrees are not elementarily equivalent. Archiv für mathematische Logik und Grundlagenforschung, vol. 24 (1984), pp. 137– 139. - Richard A. Shore. The structure of the degrees of unsolvabitity. Recursion theory, edited by Anil Nerode and Richard A. Shore, Proceedings of symposia in pure mathematics, vol. 42, American Mathematical Society, Providence1985, pp. 33– 51. - Theodore A. Slaman and W. Hugh Woodin. Definability in the Turing degrees. Illinois journal of mathematics, vol. 30 (1986), pp. 320– 334." Journal of Symbolic Logic 55, no. 1 (March 1990): 358–60. http://dx.doi.org/10.2307/2274995.

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Becker, Howard S. "R. Dougherty and A. S. Kechris. Hausdorff measures and sets of uniqueness for trigonometric series. Proceedings of the American Mathematical Society, vol. 105 (1989), pp. 894–897. - Alexander S. Kechris and Alain Louveau. Covering theorems for uniqueness and extended uniqueness sets. Colloquium mathematicum, vol. 59 (1990), pp. 63–79. - Alexander S. Kechris. Hereditary properties of the class of closed sets of uniqueness for trigonometric series. Israel journal of mathematics, vol. 73 (1991), pp. 189–198. - A. S. Kechris and A. Louveau. Descriptive set theory and harmonic analysis. The journal of symbolic logic, vol. 57 (1992), pp. 413–441." Bulletin of Symbolic Logic 8, no. 1 (March 2002): 94–95. http://dx.doi.org/10.2178/bsl/1182353856.

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Ressayre, J. P. "Jon Barwise and John Schlipf. On recursively saturated models of arithmetic. Model theory and algebra, A memorial tribute to Abraham Robinson, edited by D. H. Saracino and V. B. Weispfenning, Lecture notes in mathematics, vol. 498, Springer-Verlag, Berlin, Heidelberg, and New York, 1975, pp. 42–55. - Patrick Cegielski, Kenneth McAloon, and George Wilmers. Modèles récursivement saturés de l'addition et de la multiplication des entiers naturels. Logic Colloquium '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and London, 1982, pp. 57–68. - Julia F. Knight. Theories whose resplendent models are homogeneous. Israel journal of mathematics, vol. 42 (1982), pp. 151–161. - Julia Knight and Mark Nadel. Expansions of models and Turing degrees. The journal of symbolic logic, vol. 47 (1982), pp. 587–604. - Julia Knight and Mark Nadel. Models of arithmetic and closed ideals. The journal of symbolic logic, vol. 47 no. 4 (for 1982, pub. 1983), pp. 833–840. - Henryk Kotlarski. On elementary cuts in models of arithmetic. Fundamenta mathematicae, vol. 115 (1983), pp. 27–31. - H. Kotlarski, S. Krajewski, and A. H. Lachlan. Construction of satisfaction classes for nonstandard models. Canadian mathematical bulletin—Bulletin canadien de mathématiques, vol. 24 (1981), pp. 283–293. - A. H. Lachlan. Full satisfaction classes and recursive saturation. Canadian mathematical bulletin—Bulletin canadien de mathématiques, pp. 295–297. - Leonard Lipshitz and Mark Nadel. The additive structure of models of arithmetic. Proceedings of the American Mathematical Society, vol. 68 (1978), pp. 331–336. - Mark Nadel. On a problem of MacDowell and Specker. The journal of symbolic logic, vol. 45 (1980), pp. 612–622. - C. Smoryński. Back-and-forth inside a recursively saturated model of arithmetic. Logic Colloquium '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and London, 1982, pp. 273–278. - C. Smoryński and J. Stavi. Cofinal extension preserves recursive saturation. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7,1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 338–345. - George Wilmers. Minimally saturated models. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7, 1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 370–380." Journal of Symbolic Logic 52, no. 1 (March 1987): 279–84. http://dx.doi.org/10.2307/2273884.

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Cherlin, Gregory L. "Angus Macintyre, Kenneth McKenna, and Lou van den Dries. Elimination of quantifiers in algebraic structures. Advances in mathematics, vol. 47 (1983), pp. 74–87. - L. P. D. van den Dries. A linearly ordered ring whose theory admits elimination of quantifiers is a real closed field. Proceedings of the American Mathematical Society, vol. 79 (1980), pp. 97–100. - Bruce I. Rose. Rings which admit elimination of quantifiers. The journal of symbolic logic, vol. 43 (1978), pp. 92–112; Corrigendum, vol. 44 (1979), pp. 109–110. - Chantal Berline. Rings which admit elimination of quantifiers. The journal of symbolic logic, vol. 43 (1978), vol. 46 (1981), pp. 56–58. - M. Boffa, A. Macintyre, and F. Point. The quantifier elimination problem for rings without nilpotent elements and for semi-simple rings. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7, 1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 20–30. - Chantal Berline. Elimination of quantifiers for non semi-simple rings of characteristic p. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7, 1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 10–19." Journal of Symbolic Logic 50, no. 4 (December 1985): 1079–80. http://dx.doi.org/10.2307/2273998.

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Baumgartner, James E. "Edwin W. Miller. On a property of families of sets. English with Polish summary. Sprawozdania z posiedzeń Towarzystwa Naukowego Warszawskiego (Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie), Class III, vol. 30 (1937), pp. 31–38. - Ben Dushnik and Miller E. W.. Partially ordered sets. American journal of mathematics, vol. 63 (1941), pp. 600–610. - P. Erdős. Some set-theoretical properties of graphs. Revista, Universidad Nacional de Tucumán, Serie A, Matemáticas y física teórica, vol. 3 (1942), pp. 363–367. - G. Fodor. Proof of a conjecture of P. Erdős. Acta scientiarum mathematicarum, vol. 14 no. 4 (1952), pp. 219–227. - P. Erdős and Rado R.. A partition calculus in set theory. Bulletin of the American Mathematical Society, vol. 62 (1956), pp. 427–489. - P. Erdős and Rado R.. Intersection theorems for systems of sets. The journal of the London Mathematical Society, vol. 35 (1960), pp. 85–90. - A. Hajnal. Some results and problems on set theory. Acta mathematica Academiae Scientiarum Hungaricae, vol. 11 (1960), pp. 277–298. - P. Erdős and Hajnal A.. On a property of families of sets. Acta mathematica Academiae Scientiarum Hungaricae, vol. 12 (1961), pp. 87–123. - A. Hajnal. Proof of a conjecture of S. Ruziewicz. Fundamenta mathematicae, vol. 50 (1961), pp. 123–128. - P. Erdős, Hajnal A. and Rado R.. Partition relations for cardinal numbers. Acta mathematica Academiae Scientiarum Hungaricae, vol. 16 (1965), pp. 93–196. - P. Erdős and Hajnal A.. On a problem of B. Jónsson. Bulletin de l'Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques et physiques, vol. 14 (1966), pp. 19–23. - P. Erdős and Hajnal A.. On chromatic number of graphs and set-systems. Acta mathematica Academiae Scientiarum Hungaricae, vol. 17 (1966), pp. 61–99." Journal of Symbolic Logic 60, no. 2 (June 1995): 698–701. http://dx.doi.org/10.2307/2275868.

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Yaacov, Itaï Ben. "Ilijas Farah, Bradd Hart, and David Sherman. Model theory of operator algebras I: stability. Bulletin of the London Mathematical Society, vol. 45 (2013), no. 4, pp. 825–838, doi:10.1112/blms/bdt014. - Ilijas Farah, Bradd Hart, and David Sherman. Model theory of operator algebras II: model theory. Israel Journal of Mathematics, vol. 201 (2014), no. 1, pp. 477–505, doi:10.1007/s11856-014-1046-7. - Ilijas Farah, Bradd Hart, and David Sherman. Model theory of operator algebras III: elementary equivalence and II1factors. Bulletin of the London Mathematical Society, vol. 46 (2014), no. 3, pp. 609–628, doi:10.1112/blms/bdu012. - Isaac Goldbring, Bradd Hart, and Thomas Sinclair. The theory of tracial von Neumann algebras does not have a model companion. Journal of Symbolic Logic, vol. 78 (2013), no. 3, pp. 1000–1004." Bulletin of Symbolic Logic 21, no. 4 (December 2015): 425–27. http://dx.doi.org/10.1017/bsl.2015.32.

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Louveau, Alain. "Jack H. Silver. Counting the number of equivalence classes of Borel and coanalytic equivalence relations. Annals of mathematical logic, vol. 18 (1980), pp. 1–28. - John P. Burgess. Equivalences generated by families of Borel sets. Proceedings of the American Mathematical Society. vol. 69 (1978), pp. 323–326. - John P. Burgess. A reflection phenomenon in descriptive set theory. Fundamenta mathematicae. vol. 104 (1979), pp. 127–139. - L. Harrington and R. Sami. Equivalence relations, projective and beyond. Logic Colloquium '78, Proceedings of the Colloquium held in Mons, August 1978, edited by Maurice Boffa, Dirk van Dalen, and Kenneth McAloon, Studies in logic and the foundations of mathematics, vol. 97, North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1979, pp. 247–264. - Leo Harrington and Saharon Shelah. Counting equivalence classes for co-κ-Souslin equivalence relations. Logic Colloquium '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1982, pp. 147–152. - Jacques Stern. On Lusin's restricted continuum problem. Annals of mathematics, ser. 2 vol. 120 (1984), pp. 7–37." Journal of Symbolic Logic 52, no. 3 (September 1987): 869–70. http://dx.doi.org/10.1017/s0022481200029856.

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Bencivenga, Ermanno. "Hugues Leblanc. Preface. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. ix–x. - Hugues Leblanc. Introduction. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 3–16. - Hugues Leblanc and T. Hailperin. Non-designating singular terms. A revised reprint of XXV 87. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 17–21. - Hugues Leblanc and R. H. Thomason. Completeness theorems for some presupposition-free logics. A revised reprint of XXXVII 424. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 22–57. - Hugues Leblanc and R. K. Meyer. On prefacing (∀x) ⊃ A(Y/X) with (∀Y): a free quantification theory without identity. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 58–75. (Reprinted with revisions from Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 16 (1970), pp. 447–462. - Hugues Leblanc. Truth-value semantics for a logic of existence. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 76–90. (Reprinted with revisions from Notre Dame journal of formal logic, vol. 12 (1971), pp. 153–168.) - Hugues Leblanc and R. K. Meyer. Open formulas and the empty domain. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 91–98. (Reprinted from Archiv für mathematische Logik und Grundlagenforschung, vol. 12 (1969), pp. 78–84.) - K. Lambert, Hugues Leblanc, and R. K. Meyer. A liberated version of S5. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 99–102. (Reprinted with revisions from Archiv für mathematische Logik und Grundlagenforschung, vol. 12 (1969), pp. 151–154.) - Hugues Leblanc. On dispensing with things and worlds. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 103–119. (Reprinted with revisions from Logic and ontology, edited by Milton K. Munitz, New York University Press, New York 1973, pp. 241–259.) - Hugues Leblanc. Introduction. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 123–138. - Hugues Leblanc. A simplified account of validity and implication for quantificational logic. A revised reprint of XXXV 466. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 139–143. - Hugues Leblanc. A simplified strong completeness proof for QC=. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 144–155. (Reprinted with minor revisions from Akten des XIV. Internationalen Kongresses für Philosophie Wien, 2.-9. September 1968, vol. 3, Logik Erkenntnis- und Wissenschaftstheorie Sprachphilosophie Ontologie und Metaphysik, Universität Wien, Herder, Vienna 1969, pp. 83–96.) - Hugues Leblanc. Truth-value assignments and their cardinality. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 156–165. (Reprinted with revisions from Philosophia, vol. 7 (1978), pp. 305–316.) - Hugues Leblanc. Three generalizations of a theorem of Beth's. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 166–176. (Reprinted with revisions from Logique et analyse, n.s. vol. 12 (1969), pp. 205–220.) - Hugues Leblanc and R. K. Meyer. Truth-value semantics for the theory of types. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 177–197. (Reprinted with revisions from Philosophical problems in logic, Some recent developments, edited by Karel Lambert, Synthese library, D. Reidel Publishing Company, Dordrecht 1970, pp. 77–101.) - Hugues Leblanc. Wittgenstein and the truth-functionality thesis. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 198–204. (Reprinted with revisions from American philosophical quarterly, vol. 9 (1972), pp. 271–274.) - Hugues Leblanc. Matters of relevance. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 205–219. (Reprinted from Journal of philosophical logic, vol. 1 (1972), pp. 269–286. Also reprinted in Exact philosophy, Problems, tools, and goals, edited by Mario Bunge, Synthese library, D. Reidel Publishing Company, Dordrecht and Boston 1973, pp. 3–20.) - Hugues Leblanc and G. Weaver. Truth-functionality and the ramified theory of types. A revised reprint of XLII 313. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 220–235. - Hugues Leblanc. That Principia mathematica, first edition, has a predicative interpretation after all. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 236–239. (Reprinted with revisions from Journal of philosophical logic, vol. 4 (1975), pp. 67–70.) - H. Goldberg, Hugues Leblanc, and G. Weaver. A strong completeness theorem for three-valued logic: part I. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 240–246. (Reprinted with revisions from Notre Dame journal of formal logic, vol. 15 (1974), pp. 325–330.) - Hugues Leblanc. A strong completeness theorem for three-valued logic: part II. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 247–257. (Reprinted with revisions from Notre Dame journal of formal logic, vol. 18 (1977), pp. 107–116.) - Hugues Leblanc and R. P. McArthur. A completeness result for quantificational tense logic. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 258–266. (Reprinted with revisions from Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 22 (1976), pp. 89–96.) - Hugues Leblanc. Semantic deviations. A revised reprint of XLII 313. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 267–280. - Hugues Leblanc. Introduction. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 283–292. - Hugues Leblanc. Marginalia on Gentzen's Sequenzen-Kalkulë. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 293–300. (Reprinted with revisions from Contributions to logic and methodology in honor of J. M. Bocheński, edited by Anna-Teresa Tymieniecka in collaboration with Charles Parsons, North-Holland Publishing Company, Amsterdam 1965, pp. 73–83.) - Hugues Leblanc. Structural rules of inference. A revised reprint of XXVIII 256. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 301–305. - Hugues Leblanc. Proof routines for the propositional calculus. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 306–327. (Reprinted with revisions from Notre Dame journal of formal logic, vol. 4 (1963), pp. 81–104.) - Hugues Leblanc. Two separation theorems for natural deduction. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 328–349. (Reprinted with revisions from Notre Dame journal of formal logic, vol. 7 (1966), pp. 159–180.) - Hugues Leblanc. Two shortcomings of natural deduction. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 350–357. (Reprinted with revisions from The journal of philosophy, vol. 63 (1966), pp. 29–37.) - Hugues Leblanc. Subformula theorems for N-sequents. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 358–381. (Reprinted with minor revisions from The journal of symbolic logic, vol. 33 (1968), pp. 161–179.) - E. W. Beth and Hugues Leblanc. A note on the intuitionist and the classical propositional calculus. A revised reprint of XXV 351. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 382–384. - Hugues Leblanc and N. D. Belnap Jr. Intuitionism reconsidered. A revised reprint of XXVIII 256. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 385–389. - N. D. BelnapJr., Hugues Leblanc, and R. H. Thomason. On not strengthening intuitionistic logic. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 390–396. (Reprinted with revisions from Notre Dame journal of formal logic, vol. 4 no. 4 (for 1963, pub. 1964), pp. 313–320.) - Hugues Leblanc and R. H. Thomason. The demarcation line between intuitionist logic and classical logic. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 397–403. (Reprinted with revisions from Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 257–262.) - Hugues Leblanc. Boolean algebra and the propositional calculus. A revised reprint of XXXVII 755. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 404–407. - Hugues Leblanc. The algebra of logic and the theory of deduction. A revised reprint of XXXVII 755. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 408–413. - Hugues Leblanc and R. H. Thomason. All or none: a novel choice of primitives for elementary logic. A revised reprint of XXXIV 124. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 414–421. - Hugues Leblanc and R. K. Meyer. Matters of separation. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 422–430. (Reprinted with revisions from Notre Dame journal of formal logic, vol. 13 (1972), pp. 229–236.) - Hugues Leblanc. Generalization in first-order logic. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 431–452. (Reprinted with revisions from Notre Dame journal of formal logic, vol. 20 (1979), pp. 835–857.)." Journal of Symbolic Logic 50, no. 1 (March 1985): 227–31. http://dx.doi.org/10.2307/2273801.

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"IVAN SLESHYNSKY AS A POPULARIZER OF THE IDEAS OF MATHEMATICAL LOGIC IN UKRAINE." Journal of V. N. Karazin Kharkiv National University, Series "Philosophy. Philosophical Peripeteias", no. 62 (2020): 99–107. http://dx.doi.org/10.26565/2226-0994-2020-62-11.

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The first half of the twentieth century was marked by the simultaneous development of logic and mathematics. Logic offered the necessary means to justify the foundations of mathematics and to solve the crisis that arose in mathematics in the early twentieth century. In European science in the late nineteenth century, the ideas of symbolic logic, based on the works of J. Bull, S. Jevons and continued by C. Pierce in the United States and E. Schroeder in Germany were getting popular. The works by G. Frege and B. Russell should be considered more progressive towards the development of mathematical logic. The perspective of mathematical logic in solving the crisis of mathematics in Ukraine was noticed by Professor of Mathematics of Novorossiysk (Odesa) University Ivan Vladislavovich Sleshynsky. Sleshynsky (1854 –1931) is a Doctor of Mathematical Sciences (1893), Professor (1898) of Novorossiysk (Odesa) University. After studying at the University for two years he was a Fellow at the Department of Mathematics of Novorossiysk University, defended his master’s thesis and was sent to a scientific internship in Berlin (1881–1882), where he listened to the lectures by K. Weierstrass, L. Kronecker, E. Kummer, G. Bruns. Under the direction of K. Weierstrass he prepared a doctoral dissertation for defense. He returned to his native university in 1882, and at the same time he was a teacher of mathematics in the seminary (1882–1886), Odesa high schools (1882–1892), and taught mathematics at the Odesa Higher Women’s Courses. Having considerable achievements in the field of mathematics, in particular, Pringsheim’s Theorem (1889) proved by Sleshinsky on the conditions of convergence of continuous fractions, I. Sleshynsky drew attention to a new direction of logical science. The most significant work for the development of national mathematical logic is the translation by I. Sleshynsky from the French language “Algebra of Logic” by L. Couturat (1909). Among the most famous students of I. Sleshynsky, who studied and worked at Novorossiysk University and influenced the development of mathematical logic, one should mention E. Bunitsky and S. Shatunovsky. The second period of scientific work of I. Sleshynsky is connected with Poland. In 1911 he was invited to teach mathematical disciplines at Jagiellonian University and focused on mathematical logic. I. Sleshynsky’s report “On Traditional Logic”, delivered at the meeting of the Philosophical Society in Krakow. He developed the common belief among mathematicians that logic was not necessary for mathematics. His own experience of teaching one of the most difficult topics in higher mathematics – differential calculus, pushed him to explore logic, since the requirement of perfect mathematical proof required this. In one of his further works of this period, he noted the promising development of mathematical logic and its importance for mathematics. He claimed that for the mathematics of future he needed a new logic, which he saw in the “Principles of Mathematics” by A. Whitehead and B. Russell. Works on mathematical logic by I. Sleszynski prompted many of his students in Poland to undertake in-depth studies in this field, including T. Kotarbiński, S. Jaśkowski, V. Boreyko, and S. Zaremba. Thanks to S. Zaremba, I. Sleshynsky managed to complete the long-planned concept, a two-volume work “Theory of Proof” (1925–1929), the basis of which were lectures of Professor. The crisis period in mathematics of the early twentieth century, marked by the search for greater clarity in the very foundations of mathematical reasoning, led to the transition from the study of mathematical objects to the study of structures. The most successful means of doing this were proposed by mathematical logic. Thanks to Professor I. Sleshynsky, who succeeded in making Novorossiysk (Odesa) University a center of popularization of mathematical logic in the beginning of the twentieth century the ideas of mathematical logic in scientific environment became more popular. However, historical events prevented the ideas of mathematical logic in the domestic scientific space from the further development.
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Buss, Samuel, Helmut Schwichtenberg, and Ulrich Kohlenbach. "Mathematical Logic: Proof Theory, Constructive Mathematics." Oberwolfach Reports, 2008, 907–52. http://dx.doi.org/10.4171/owr/2008/18.

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Schwichtenberg, Helmut, Vladimir Keilis-Borok, and Samuel Buss. "Mathematical Logic: Proof Theory, Type Theory and Constructive Mathematics." Oberwolfach Reports, 2005, 779–813. http://dx.doi.org/10.4171/owr/2005/14.

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SIEG, WILFRIED, and PATRICK WALSH. "NATURAL FORMALIZATION: DERIVING THE CANTOR-BERNSTEIN THEOREM IN ZF." Review of Symbolic Logic, November 18, 2019, 1–35. http://dx.doi.org/10.1017/s175502031900056x.

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Abstract Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas. To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at http://www.phil.cmu.edu/legacy/Proof_Site/. We must—that is my conviction—take the concept of the specifically mathematical proof as an object of investigation. Hilbert 1918
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Sümmermann, Moritz Lucius, Daniel Sommerhoff, and Benjamin Rott. "Mathematics in the Digital Age: The Case of Simulation-Based Proofs." International Journal of Research in Undergraduate Mathematics Education, February 15, 2021. http://dx.doi.org/10.1007/s40753-020-00125-6.

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AbstractDigital transformation has made possible the implementation of environments in which mathematics can be experienced in interplay with the computer. Examples are dynamic geometry environments or interactive computational environments, for example GeoGebra or Jupyter Notebook, respectively. We argue that a new possibility to construct and experience proofs arises alongside this development, as it enables the construction of environments capable of not only showing predefined animations, but actually allowing user interaction with mathematical objects and in this way supporting the construction of proofs. We precisely define such environments and call them “mathematical simulations.” Following a theoretical dissection of possible user interaction with these mathematical simulations, we categorize them in relation to other environments supporting the construction of mathematical proofs along the dimensions of “interactivity” and “formality.” Furthermore, we give an analysis of the functions of proofs that can be satisfied by simulation-based proofs. Finally, we provide examples of simulation-based proofs in Ariadne, a mathematical simulation for topology. The results of the analysis show that simulation-based proofs can in theory yield most functions of traditional symbolic proofs, showing promise for the consideration of simulation-based proofs as an alternative form of proof, as well as their use in this regard in education as well as in research. While a theoretical analysis can provide arguments for the possible functions of proof, they can fulfil their actual use and, in particular, their acceptance is of course subject to the sociomathematical norms of the respective communities and will be decided in the future.
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Strauss, Daniel Francois. "The Fall and Original Sin of Set Theory." Phronimon 19 (January 10, 2019). http://dx.doi.org/10.25159/2413-3086/4983.

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Hermann Weyl published a brief survey as preface to a review of The Philosophy of Bertrand Russell in 1946. In this survey he used the phrase, “The Fall and Original Sin of Set Theory.” Investigating the background of this remark will require that we pay attention to a number of issues within the foundations of mathematics. For example: Did God make the integers—as Kronecker alleged? Is mathematics set theory? Attention will also be given to axiomatic set theory and relevant ontic pre-conditions, such as the difference between number and number symbols, to number as “an aspect of objective reality” (Gödel), integers and induction (Skolem) as well as to the question if infinity—as endlessness—could be completed. In 1831 Gauss objected to viewing the infinite as something completed, which is not allowed in mathematics. It will be argued that the actual infinite is rather connected to what is present “at once,” as an infinite totality. By the year 1900 mathematicians believed that mathematics had reached absolute rigour, but unfortunately the rest of the twentieth century witnessed the opposite. The axiom of infinity ruined the expectations of logicism—mathematics cannot be reduced to logic. The intuitionism of Brouwer, Weyl and others launched a devastating attack on classical analysis, further inspired by the outcome of Gödel’s famous proof of 1931, in which he has shown that a formal mathematical system is inconsistent or incomplete. Intuitionism created a whole new mathematics, which finds no counter-part in classical mathematics. Slater remarked that within this logical paradise of Russell lurked a serpent, hidden behind the unjustified employment of the at once infinite. According to Weyl, “This is the Fall and original sin of set theory for which it is justly punished by the antinomies.” In conclusion, a few systematic distinctions are introduced.
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Kusraev, A. G., and S. S. Kutateladze. "Приглашение в булевозначный анализ." Владикавказский математический журнал, no. 2 (July 4, 2018). http://dx.doi.org/10.23671/vnc.2018.2.14723.

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This is a short invitation to the field of Boolean valued analysis. Model theory evaluates and counts truth and proof. The chase of truth not only leads us close to the truth we pursue but also enables us to nearly catch up with many other instances of truth which we were not aware nor even foresaw at the start of~the rally pursuit. That is what we have learned from Boolean valued models of set theory. These models stem from the famous works by Paul Cohen on the continuum hypothesis. They belong to logic and yield a~profusion of the surprising and unforeseen visualizations of the ingredients of mathematics. Many promising opportunities are open to modeling the powerful habits of reasoning and verification. Boolean valued analysis is a blending of analysis and Boolean valued models. Adaptation of the ideas of Boolean valued models to functional analysis projects among the most important directions of developing the synthetic methods of mathematics. This approach yields the new models of numbers, spaces, and types of equations. The content expands of all available theorems and algorithms. The whole methodology of mathematical research is enriched and renewed, opening up absolutely fantastic opportunities. We can now transform matrices into numbers, embed function spaces into a straight line, yet having still uncharted vast territories of new knowledge. The article advertised two books that crown our thought about and research into the field.
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Barnet, Belinda. "Machinic Heterogenesis and Evolution." M/C Journal 2, no. 6 (September 1, 1999). http://dx.doi.org/10.5204/mcj.1789.

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"I write for a species that does not yet exist." -- Nietzsche (958) III. Note on Self-Organisation and Selectionism According to your mainstream brand neo-Darwinian biologist, natural selection is the stuff of which evolution is made, the First Principle of life. There is nothing in the natural world which cannot be explained by random mutations within the genome and subsequent selection of the fittest form by the natural environment. Beyond the constraints set by the period of waiting for mutations to occur and external conditions, there are no limits to this system, and an organism forms from scratch to a furry crawling thing in a gradual process reliant on external factors. Neo-Darwinism is an attempt to reconcile two theories which are quite simply at odds with one another: Mendelian genetics, which claims that organisms do not change with time, and Darwinism, which claims that they do. This is usually done in a mathematical way, with natural selection as the linchpin of some tight equations. There can be no internal feedback from the body (phenotype) to the genes (genotype). There is no self-organising adaptive order: all emerges from the process of selection as a carefully articulated tree diagram, and adapts over eons. As the Darwinian critic Arthur Koestler pointed out, natural selection is hence the only process found in nature which is devoid of feedback. Neo-Darwinian theory is both unfalsifiable and all-pervasive; it is easy to forget that it is a theory which has not yet been proven beyond doubt by paleontological fact, and that Darwin himself suggested there may be processes other than natural selection at work in the unfolding of life. There are a couple of rogue biologists and a-life crazies, however, that don't believe the Selectionist hype. They are not suggesting that natural selection is a dud theory, but simply that there might be other factors involved, and that the really interesting questions don't just concern life as a Darwinian competition between furry, crawling things, but the interplay between structure and chaos at the basic levels of the system which might give rise to it. Biologists such as Brian Goodwin and Stuart Kauffman take issue with this, claiming that an understanding of life should begin at a more fundamental level than tree diagrams and zoology -- molecular biology, biochemistry, complexity theory. This is the 'language' of life: the way that structure spontaneously emerges from chaos. Niles Eldredge and Stephen Jay Gould looked at the fossil record a few years back and decided that there is no proof that one species turns into another slowly: the mathematics of the Neo-Darwinists relied upon the idea that species took hundreds of millions of years to evolve eyes and ears and legs and wings, branching off into other species in the manner of a tree diagram over billions of years. What Eldredge and Gould found was that species seem to spontaneously emerge fully formed: there is minimal variation going on. A species emerges rapidly, it lasts for a time (often a short time), and then it dies off. The in-between period, the period of mutation and selectionism, is largely unaccounted for by the fossil record, especially considering the importance of such transitory phases to the neo-Darwinists. There are many 'missing links' in the record. For over twenty years, Stuart Kauffman has been going on about what we might call a Second Principle in evolutionary biology: self-organisation. He argues that because natural selection alone is not enough to explain the relatively short timescale on which life arose, some other ordering principle is necessary. He locates this, as Katherine Hayles observes, in the ability of complex systems to self-organise (241). A self-organising system involves the heresy of internal feedback and internally-produced constraints. Living creatures would converge upon certain forms as much as diverge from them due to the influence of mutations caused by cosmic rays, wild chance and external factors. Creatures will not just evolve over billions of years due to selection, but will appear in a more concerted and spontaneous manner. Systems will seek their own order. The heresy in this (as far as neo-Darwinians are concerned, but not all evolutionary biologists) is located in the fact that such enabling constraints emerge from within the system itself. Consequently, natural selection is not the only force at work in evolution. The system is its own material of expression, and can generate its own tendencies and limits. Kauffman calls this process antichaos, or "order for free" (335). One can sense that such a theory would be objectionable to biologists: there is nothing distinctively biological about this explanation, which in fact borrows from physics and complexity theory, and it explores living organisms, chemical compositions and non-biological aggregates alike as systems, privileging no particular machine. A 'complex system', in particular, can be anything from the stockmarket to a flock of birds. XI. Sonicform We might note here a similarity with virtual artist Keith Nettos's Java-based sound system, Sonicform, whose evolving sound structures can be obtained from the artist on request <lucidweave@usa.net>: the divide between living and non-living is not the issue. As Keith puts it, "it's an echo of that Descartian dichotomy between mind and matter. Do such distinctions help us to know ourselves better? I'm not sure that they do". Sonicform is more a world of Newtonian discovery than biblical creation. Self-organisation works on a generative systemic level, and is a prerequisite more so than a defining quality of life or evolution; it is necessary but not sufficient to characterise an organic system. The computer is the perfect environment in which to explore the confusion and commonalities between animate and inanimate systems, and in that confusion, reveal something of the processes underlying the actual generation of self and order in the universe. Information-processing, and life, require a certain type of complexity. The system must be dynamic, yet allow for novel patterns. The computer emphasises the logic as well as the mechanics of life, which are then honed and honoured by the more familiar conception of natural selection. IV. Self-organisation is the natural consequence of simple components (cells, units of sound, air molecules, genes) interacting via equally simple rules. Patterns and forms emerge from the collective raucous, and these forms give rise to other forms. The components in such a system are bimbos: they have no idea what is going on in the greater body, and don't care. In other words, a complex system emerges from lots of small but well-chosen components interacting in a rule-governed way, developing a larger behaviour or pattern which cannot be predicted or divined from these constituent parts. Random mutation and selection will act upon such a system -- this is how Selectionism fits in: forms will not just evolve from scratch via selection, but will spontaneously emerge from within the system, working in conjunction with the First Principle. IV. Sonicform In the Sonicform system, the components are 'sound fragments', the samples attached to the images in the top left-hand corner of the screen at startup, and also the people seated at terminals who interact with these fragments. Although it might seem to be stretching the concept of systemic components to include the user population, the fact that the emerging pattern is dependent on these users to evolve renders them part of the system. The organisation of a machine has less to do with its materiality than with the inter-relations of its components. The rules in Sonicform are the 'sound controllers' located on the right-hand side of the screen, containing basic instructions such as "play sound", "loop sound" and "stop sound" that control the sound fragments and consequently limit the structure of the emerging acoustic pattern. Because Sonicform is linked via the Net to 'sonicserver' and consequently the multiple versions of itself which are being executed at any point in time, any changes that a user makes (e.g. attraction towards a particular kind of sound) are detected by sonicserver and fed back into a primary chain structure. This is the formative basis of Sonicform's 'evolution': a selection of internal behavioural constraints generated by its constituent parts. The heresy in this is the implication that both biological and technical systems are capable of self-organisation and evolution, that both are constellations of universes which are capable of autonomy and complexity (and 'life' as a certain form of complexity). This is not anti-humanist. It's not even post-humanist. Ideology is a human concept which is brought to bear on technology. We're talking a different register altogether. Technical machines, organic machines, conceptual machines: each will beget the other, each will inscribe its own pattern on the process, each will redefine the limits of such connections. VII. The Death of Metaphor: All That Consists Is Real 'Machinic heterogenesis', a term used by Felix Guattari in his book Chaosmosis, is a mode of being and production that draws on complexity theory and the work of Francisco Varela (a biologist interested in self-organisation in immune networks) and Kauffman. Guattari extends the concept of self-organisation to create a pragmatic philosophy. Machinic heterogenesis is a term to describe the way that the machines which populate the universe connect with each other, mutually affect each other, exchange segments and then bifurcate into new machines. Collective existential mutation. When we sit at a computer screen, we are connected with the computer's universes of reference through the circuits of sight, the play of fingers across the keyboard, the conceptual and logical limits of the exchange laid down by both parties. There is a certain synchrony going on across the zone of intersection and compromise to the limits of this exchange. In other words, the limits of the medium define the exchange and what we are becoming as we connect with it. What is the 'ness' of the computer medium, and what are the possible universes of exchange which extend from this? Sonicform explores this exchange through sound, and through a system which explicitly invites us to be a part of an evolving structure. The use of complexity theory and evolution in Sonicform makes explicit the rethinking of machines which we have been doing here in general: machines speak to machines before they speak to Man, and the ontological domains that they reveal and secrete tend towards pattern in an innate way, determined by the mode of aggregation of their constituent parts. Sonicform rethinks technology in terms of evolutionary, collective entities. And this rethinking allows for the particular qualities of the medium itself, its own characteristics, its own unique interpretations of our model of evolution, to express themselves. Here I might note something: evolution cannot be naturalised and reified as an entity independent of the conceptual, technical and scientific machinery of its production. In the eagerness to import biological models to the computer in a-life, we sometimes forget that from its very origins, the human species has been constituted by technical evolution, and that it is the mediation afforded by technics which makes "it impossible simply to describe evolution in terms of a self-contained, or monadic, subject that passively 'adapts' to an object-like environment" (Pearson 4). Similarly, we have produced our various models of evolution by analysing the 'natural environment' through the mediation of technology. Technology has always enjoyed more than just the position of a neutral tool to locate and test Nature, and has its own unique limits and qualities to contribute to anything we produce with it. So this will be the beginning of our rethinking. Constellations of universes colliding, machines exchanging particularities, components that retain their autonomy and yet can collect and self-organise into complex systems, even life. "The ideas that we have been devoting space to here -- instability, fluctuation, complex systems -- diffuse into the social sciences", in the words of Ilya Prigogine (312). If we can create an evolving complex system on the screen which we ourselves are components of, we tend to rethink the interface between nature and technology. What does it say about the "reference point" of the natural world when creatures whose entire function consists of weird acoustic dances across computer circuitry begin to self-replicate and exhibit the signs of open-ended evolution, resulting in formations which no longer have analogues in the 'natural' world? I'd like to hesitate a start here. Biology is its own material of semiotic expression. Techné is its own material of semiotic expression. To address the interface between nature and technology, we need a philosophy of cells, flocks, patterns, components, motors, and elements. We need a philosophy that will create an interference pattern across the zone of intersection. References Hayles, N. Katherine. How We Became Posthuman: Virtual Bodies in Cybernetics, Literature and Informatics. Chicago: U of Chicago P, 1999. Kauffman, Stuart. "Order For Free." The Third Culture. Ed. Brockman. New York: Touchstone Press. Nietzsche, Friedrich. The Will to Power. Trans. W. Kaufmann and R. J. Hollingdale. New York: Random House, 1968. Prigogine, Ilya, and Isabelle Stengers. Order Out of Chaos: Man's New Dialogue with Nature. London: Bantam Books, 1984. Pearson, Keith Ansell. Viroid Life. London: Routledge, 1997. This work is part of the Australian Network for Art and Technology's Deep Immersion: Creative Collaborations project, funded by the AFC. Citation reference for this article MLA style: Belinda Barnet. "Machinic Heterogenesis and Evolution: Collected Notes on Sound, Machines and Sonicform." M/C: A Journal of Media and Culture 2.6 (1999). [your date of access] <http://www.uq.edu.au/mc/9909/sonic.php>. Chicago style: Belinda Barnet, "Machinic Heterogenesis and Evolution: Collected Notes on Sound, Machines and Sonicform," M/C: A Journal of Media and Culture 2, no. 6 (1999), <http://www.uq.edu.au/mc/9906/sonic.php> ([your date of access]). APA style: Belinda Barnet. (1999) Machinic heterogenesis and evolution: collected notes on sound, machines and Sonicform. M/C: A Journal of Media and Culture 2(6). <http://www.uq.edu.au/mc/9909/sonic.php> ([your date of access]).
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