The Collected Papers of Gerhard Gentzen
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Gödel on Finitism, Constructivity and Hilbert's Program
Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on finitism, constructivity and Hilbert’s program Solomon Feferman 1. Gödel, Bernays, and Hilbert. The correspondence between Paul Bernays and Kurt Gödel is one of the most extensive in the two volumes of Gödel’s collected works devoted to his letters of (primarily) scientific, philosophical and historical interest. It ranges from 1930 to 1975 and deals with a rich body of logical and philosophical issues, including the incompleteness theorems, finitism, constructivity, set theory, the philosophy of mathematics, and post- Kantian philosophy, and contains Gödel’s thoughts on many topics that are not expressed elsewhere. In addition, it testifies to their life-long warm personal relationship. I have given a detailed synopsis of the Bernays Gödel correspondence, with explanatory background, in my introductory note to it in Vol. IV of Gödel’s Collected Works, pp. 41- 79.1 My purpose here is to focus on only one group of interrelated topics from these exchanges, namely the light that ittogether with assorted published and unpublished articles and lectures by Gödelthrows on his perennial preoccupations with the limits of finitism, its relations to constructivity, and the significance of his incompleteness theorems for Hilbert’s program.2 In that connection, this piece has an important subtext, namely the shadow of Hilbert that loomed over Gödel from the beginning to the end of his career. 1 The five volumes of Gödel’s Collected Works (1986-2003) are referred to below, respectively, as CW I, II, III, IV and V. CW I consists of the publications 1929-1936, CW II of the publications 1938-1974, CW III of unpublished essays and letters, CW IV of correspondence A-G, and CW V of correspondence H-Z. -
Lorenzen's Proof of Consistency for Elementary Number Theory [With An
Lorenzen’s proof of consistency for elementary number theory Thierry Coquand Computer science and engineering department, University of Gothenburg, Sweden, [email protected]. Stefan Neuwirth Laboratoire de mathématiques de Besançon, Université Bourgogne Franche-Comté, France, [email protected]. Abstract We present a manuscript of Paul Lorenzen that provides a proof of con- sistency for elementary number theory as an application of the construc- tion of the free countably complete pseudocomplemented semilattice over a preordered set. This manuscript rests in the Oskar-Becker-Nachlass at the Philosophisches Archiv of Universität Konstanz, file OB 5-3b-5. It has probably been written between March and May 1944. We also com- pare this proof to Gentzen’s and Novikov’s, and provide a translation of the manuscript. arXiv:2006.08996v1 [math.HO] 16 Jun 2020 Keywords: Paul Lorenzen, consistency of elementary number the- ory, free countably complete pseudocomplemented semilattice, inductive definition, ω-rule. We present a manuscript of Paul Lorenzen that arguably dates back to 1944 and provide an edition and a translation, with the kind permission of Lorenzen’s daughter, Jutta Reinhardt. It provides a constructive proof of consistency for elementary number theory by showing that it is a part of a trivially consistent cut-free calculus. The proof resorts only to the inductive definition of formulas and theorems. More precisely, Lorenzen proves the admissibility of cut by double induction, on the complexity of the cut formula and of the derivations, without using any ordinal assignment, contrary to the presentation of cut elimination in most standard texts on proof theory. -
Common Sense for Concurrency and Strong Paraconsistency Using Unstratified Inference and Reflection
Published in ArXiv http://arxiv.org/abs/0812.4852 http://commonsense.carlhewitt.info Common sense for concurrency and strong paraconsistency using unstratified inference and reflection Carl Hewitt http://carlhewitt.info This paper is dedicated to John McCarthy. Abstract Unstratified Reflection is the Norm....................................... 11 Abstraction and Reification .............................................. 11 This paper develops a strongly paraconsistent formalism (called Direct Logic™) that incorporates the mathematics of Diagonal Argument .......................................................... 12 Computer Science and allows unstratified inference and Logical Fixed Point Theorem ........................................... 12 reflection using mathematical induction for almost all of Disadvantages of stratified metatheories ........................... 12 classical logic to be used. Direct Logic allows mutual Reification Reflection ....................................................... 13 reflection among the mutually chock full of inconsistencies Incompleteness Theorem for Theories of Direct Logic ..... 14 code, documentation, and use cases of large software systems Inconsistency Theorem for Theories of Direct Logic ........ 15 thereby overcoming the limitations of the traditional Tarskian Consequences of Logically Necessary Inconsistency ........ 16 framework of stratified metatheories. Concurrency is the Norm ...................................................... 16 Gödel first formalized and proved that it is not possible -
On Proofs of the Consistency of Arithmetic
STUDIES IN LOGIC, GRAMMAR AND RHETORIC 5 (18) 2002 Roman Murawski Adam Mickiewicz University, Pozna´n ON PROOFS OF THE CONSISTENCY OF ARITHMETIC 1. The main aim and purpose of Hilbert’s programme was to defend the integrity of classical mathematics (refering to the actual infinity) by showing that it is safe and free of any inconsistencies. This problem was formulated by him for the first time in his lecture at the Second International Congress of Mathematicians held in Paris in August 1900 (cf. Hilbert, 1901). Among twenty three problems Hilbert mentioned under number 2 the problem of proving the consistency of axioms of arithmetic (under the name “arithmetic” Hilbert meant number theory and analysis). Hilbert returned to the problem of justification of mathematics in lectures and papers, especially in the twentieth 1, where he tried to describe and to explain the problem more precisely (in particular the methods allowed to be used) and simultaneously presented the partial solutions obtained by his students. Hilbert distinguished between the unproblematic, finitistic part of mathematics and the infinitistic part that needed justification. Finitistic mathematics deals with so called real sentences, which are completely meaningful because they refer only to given concrete objects. Infinitistic mathematics on the other hand deals with so called ideal sentences that contain reference to infinite totalities. It should be justified by finitistic methods – only they can give it security (Sicherheit). Hilbert proposed to base mathematics on finitistic mathematics via proof theory (Beweistheorie). It should be shown that proofs which use ideal elements in order to prove results in the real part of mathematics always yield correct results, more exactly, that (1) finitistic mathematics is conservative over finitistic mathematics with respect to real sentences and (2) the infinitistic 1 More information on this can be found for example in (Mancosu, 1998). -
Biographies.Pdf
Chapter udf Biographies bio.1 Georg Cantor his:bio:can: An early biography of Georg Cantor sec (gay-org kahn-tor) claimed that he was born and found on a ship that was sailing for Saint Petersburg, Russia, and that his parents were unknown. This, however, is not true; although he was born in Saint Petersburg in 1845. Cantor received his doctorate in mathematics at the University of Berlin in 1867. He is known for his work in set theory, and is credited with founding set theory as a distinctive research discipline. He was the first to prove that there are infinite sets of different sizes. His theo- ries, and especially his theory of infini- ties, caused much debate among mathe- maticians at the time, and his work was controversial. Cantor's religious beliefs and his Figure bio.1: Georg Cantor mathematical work were inextricably tied; he even claimed that the theory of transfinite numbers had been communicated to him directly by God. Inlater life, Cantor suffered from mental illness. Beginning in 1894, and more fre- quently towards his later years, Cantor was hospitalized. The heavy criticism of his work, including a falling out with the mathematician Leopold Kronecker, led to depression and a lack of interest in mathematics. During depressive episodes, Cantor would turn to philosophy and literature, and even published a theory that Francis Bacon was the author of Shakespeare's plays. Cantor died on January 6, 1918, in a sanatorium in Halle. 1 Further Reading For full biographies of Cantor, see Dauben(1990) and Grattan-Guinness(1971). -
Gerhard Gentzen
bio.1 Gerhard Gentzen his:bio:gen: Gerhard Gentzen is known primarily sec as the creator of structural proof the- ory, and specifically the creation of the natural deduction and sequent calculus derivation systems. He was born on November 24, 1909 in Greifswald, Ger- many. Gerhard was homeschooled for three years before attending preparatory school, where he was behind most of his classmates in terms of education. De- spite this, he was a brilliant student and Figure 1: Gerhard Gentzen showed a strong aptitude for mathemat- ics. His interests were varied, and he, for instance, also write poems for his mother and plays for the school theatre. Gentzen began his university studies at the University of Greifswald, but moved around to G¨ottingen,Munich, and Berlin. He received his doctorate in 1933 from the University of G¨ottingenunder Hermann Weyl. (Paul Bernays supervised most of his work, but was dismissed from the university by the Nazis.) In 1934, Gentzen began work as an assistant to David Hilbert. That same year he developed the sequent calculus and natural deduction derivation systems, in his papers Untersuchungen ¨uber das logische Schließen I{II [Inves- tigations Into Logical Deduction I{II]. He proved the consistency of the Peano axioms in 1936. Gentzen's relationship with the Nazis is complicated. At the same time his mentor Bernays was forced to leave Germany, Gentzen joined the university branch of the SA, the Nazi paramilitary organization. Like many Germans, he was a member of the Nazi party. During the war, he served as a telecommu- nications officer for the air intelligence unit. -
Hilbert's Program Then And
Hilbert’s Program Then and Now Richard Zach Abstract Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic sys- tems. Although G¨odel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial suc- cesses, and generated important advances in logical theory and meta- theory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical un- derpinnings and consequences, and its subsequent development and influ- ences since the 1930s. Keywords: Hilbert’s program, philosophy of mathematics, formal- ism, finitism, proof theory, meta-mathematics, foundations of mathemat- ics, consistency proofs, axiomatic method, instrumentalism. 1 Introduction Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of G¨ottingen and elsewhere in the 1920s and 1930s. Briefly, Hilbert’s proposal called for a new foundation of mathematics based on two pillars: the axiomatic method, and finitary proof theory. Hilbert thought that by formalizing mathematics in axiomatic systems, and subsequently proving by finitary methods that these systems are consistent (i.e., do not prove contradictions), he could provide a philosophically satisfactory grounding of classical, infinitary mathematics (analysis and set theory). -
Hilbert's Program Then And
Hilbert’s Program Then and Now Richard Zach Abstract Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic sys- tems. Although G¨odel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial suc- cesses, and generated important advances in logical theory and meta- theory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical un- derpinnings and consequences, and its subsequent development and influ- ences since the 1930s. Keywords: Hilbert’s program, philosophy of mathematics, formal- ism, finitism, proof theory, meta-mathematics, foundations of mathemat- ics, consistency proofs, axiomatic method, instrumentalism. 1 Introduction Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of G¨ottingen and elsewhere in the 1920s and 1930s. Briefly, Hilbert’s proposal called for a new foundation of mathematics based on two pillars: the axiomatic method, and finitary proof theory. Hilbert thought that by formalizing mathematics in axiomatic systems, arXiv:math/0508572v1 [math.LO] 29 Aug 2005 and subsequently proving by finitary methods that these systems are consistent (i.e., do not prove contradictions), he could provide a philosophically satisfactory grounding of classical, infinitary mathematics (analysis and set theory). -
Propositional Logic 3
Cutback Natural deduction Hilbert calculi From syllogism to common sense: a tour through the logical landscape Propositional logic 3 Mehul Bhatt Oliver Kutz Thomas Schneider 8 December 2011 Thomas Schneider Propositional logic 3 Cutback Natural deduction Hilbert calculi And now . 1 What happened last time? 2 A calculus of natural deduction 3 Hilbert calculi Thomas Schneider Propositional logic 3 Cutback Natural deduction Hilbert calculi Semantic equivalence and normal forms α ≡ β if wα = wβ for all w Replacement Theorem: if α ≡ α0 then ϕ[α0/α] ≡ ϕ, i.e., if a subformula α of ϕ is replaced by α0 ≡ α in ϕ, then the resulting formula is equivalent to ϕ Negation normal form (NNF) is established by “pulling negation inwards”, interchanging ^ and _ Disjunctive normal form (DNF) of fct f is established by describing all lines with function value 1 in truth table Conjunctive normal form (CNF): analogous, dual Thomas Schneider Propositional logic 3 Cutback Natural deduction Hilbert calculi Functional completeness and duality Signature S is functional complete: every Boolean fct is represented by some fma in S Examples: f:, ^g, f:, _g, f!, 0g, f"g, f#g Counterexample: f!, ^, _g Dual formula: interchange ^ and _ Dual function: negate arguments and function value Duality theorem: If α represents f , then αδ represents f δ. Thomas Schneider Propositional logic 3 Cutback Natural deduction Hilbert calculi Tautologies etc. α is a tautology if w j= α (wα = 1) for all w α is satisfiable if w j= α for some w α is a contradiction if α is not satisfiable Satisfiability can be decided in nondeterministic polynomial time (NP) and is NP-hard. -
Gentzen's Original Consistency Proof and the Bar Theorem
Gentzen's original consistency proof and the Bar Theorem W. W. Tait∗ The story of Gentzen's original consistency proof for first-order number theory (Gentzen 1974),1 as told by Paul Bernays (Gentzen 1974), (Bernays 1970), (G¨odel 2003, Letter 69, pp. 76-79), is now familiar: Gentzen sent it off to Mathematische Annalen in August of 1935 and then withdrew it in December after receiving criticism and, in particular, the criticism that the proof used the Fan Theorem, a criticism that, as the references just cited seem to indicate, Bernays endorsed or initiated at the time but later rejected. That particular criticism is transparently false, but the argument of the paper remains nevertheless invalid from a constructive standpoint. In a letter to Bernays dated November 4, 1935, Gentzen protested this evaluation; but then, in another letter to him dated December 11, 1935, he admits that the \critical inference in my consistency proof is defective." The defect in question involves the application of proof by induction to certain trees, the `reduction trees' for sequents (see below and x1), of which it is only given that they are well-founded. No doubt because of his desire to reason ‘finitistically’, Gentzen nowhere in his paper explicitly speaks of reduction trees, only of reduction rules that would generate such trees; but the requirement of well- foundedness, that every path taken in accordance with the rule terminates, of course makes implicit reference to the tree. Gentzen attempted to avoid the induction; but as he ultimately conceded, the attempt was unsatisfactory. ∗My understanding of the philosophical background of Gentzen's work on consistency was enhanced by reading the unpublished manuscript \ On the Intuitionistic Background of Gentzen's 1936 Consistency Proof and Its Philosophical Aspects" by Yuta Takahashi. -
Propositional Logic Basics
To appear in Volume 3, Foundations of the Pure Sciences, in History and Philosophy of Science for Science Students Committed to Africa, Helen Lauer (Editor), Ghana University Press, Lagon, Ghana. Chapter 49 An Introduction to Classical Propositional Logic: Syntax, Semantics, Sequents by RAJEEV GORE´ 1 Introduction 3 2 Syntax 5 3 Semantics of Classical Propositional Logic 9 3.1 Truth Values, Interpretations, and Truth Tables . 9 3.2 Satisfiability, Validity and Logical Consequence . 16 4 The Sequent Calculus SK 23 5 Soundness and Completeness of SK 31 5.1 Soundness of SK . 32 5.2 Completeness of SK . 34 5.3 Deductions and Logical Consequence . 37 5.4 Algorithmic Aspects of SK . 38 6 Gentzen’s Original Sequent Calculus LK 39 7 Applications of Sequent Calculi 41 8 History 43 9 Concluding Remarks 45 Figure 1 from page 11. Truth Tables for the Semantics of Classical Propositional Logic Table 1(b) Table 1(a) ϕ ψ ϕ ψ ϕ ψ ϕ ψ ϕ ψ ∧ ∨ → ↔ ϕ ϕ t t t t t t ¬ t f t f f t f f f t f t f t t f f f f f t t Figure 2 from page 24. Sequent Calculus SK for Classical Propositional Logic (Id) Γ, ϕ = ϕ, ∆ ⇒ Γ = ϕ, ∆ Γ, ϕ = ∆ ( l) ⇒ ( r) ⇒ ¬ Γ, ϕ = ∆ ¬ Γ = ϕ, ∆ ¬ ⇒ ⇒ ¬ Γ, ϕ, ψ = ∆ Γ = ϕ, ∆ Γ = ψ, ∆ ( l) ⇒ ( r) ⇒ ⇒ ∧ Γ, ϕ ψ = ∆ ∧ Γ = ϕ ψ, ∆ ∧ ⇒ ⇒ ∧ Γ, ϕ = ∆ Γ, ψ = ∆ Γ = ϕ, ψ, ∆ ( l) ⇒ ⇒ ( r) ⇒ ∨ Γ, ϕ ψ = ∆ ∨ Γ = ϕ ψ, ∆ ∨ ⇒ ⇒ ∨ Γ = ϕ, ∆ Γ, ψ = ∆ Γ, ϕ = ψ, ∆ ( l) ⇒ ⇒ ( r) ⇒ → Γ, ϕ ψ = ∆ → Γ = ϕ ψ, ∆ → ⇒ ⇒ → Γ = ϕ, ψ, ∆ Γ, ϕ, ψ = ∆ ( l) ⇒ ⇒ ↔ Γ, ϕ ψ = ∆ ↔ ⇒ Γ, ϕ = ψ, ∆ Γ, ψ = ϕ, ∆ ( r) ⇒ ⇒ ↔ Γ = ϕ ψ, ∆ ⇒ ↔ 2 1 Introduction For over 2000 years we have tried to understand and model how we reason. -
Cut As Consequence
Cut as Consequence Curtis Franks Department of Philosophy, University of Notre Dame, Indiana, USA The papers where Gerhard Gentzen introduced natural deduction and sequent calculi suggest that his conception of logic differs substantially from now dominant views introduced by Hilbert, G¨odel, Tarski, and others. Specifically, (1) the definitive features of natural deduction calculi allowed Gentzen to assert that his classical system nk is complete based purely on the sort of evidence that Hilbert called ‘experimental’, and (2) the structure of the sequent calculi li and lk allowed Gentzen to conceptualize completeness as a question about the relationships among a system’s individual rules (as opposed to the relationship between a system as a whole and its ‘semantics’). Gentzen’s conception of logic is compelling in its own right. It is also of historical interest because it allows for a better understanding of the invention of natural deduction and sequent calculi. 1. Introduction In the opening remarks of 1930, Kurt G¨odeldescribed the construction and use of formal axiomatic systems as found in the work of Whitehead and Russell—the procedure of ‘initially taking certain evident propositions as axioms and deriving the theorems of logic and mathematics from these by means of some precisely formulated principles of inference in a purely formal way’—and remarked that when such a procedure is followed, the question at once arises whether the ini- tially postulated system of axioms and principles of inference is complete, that is, whether it actually suffices for the derivation of every logico-mathematical proposition, or whether, perhaps, it is conceivable that there are true [wahre1] propositions .