CALIJ:i'ORNIA STATE UNIVERSITY, NORTHRIDGE GODEL'S " "ON FORMALLY UNDECIDABLE PROPOSITIONS ••• " • A thesis submitted in partial satisfaction of the requirements for the Degree of Master of Sciences in Mathematics, Option II by Mary Margaret Pepe June, 1976 Pepe is approved: Date ' Dr. Donald H. Potts Date yalifornia State University, Northridge ii PREFACE Any formal system of arithmetic encompassing the addition and multi­ plication of positive integers and zero contains arithmetical proposi­ :tions which can neither be proved or disproved within the system. Kurt Godel, a young Austrian mathematician who had imigrated to the United States and became a member of the Institute for Advanced Study at Princeton, announced his discovery to the Vienna Academy of Sciences ·in 1930. He published his detailed proof in his paper, "Uber formal ,unentscheidbare Satze der Principia Mathematica und verwandter Systeme ,I," in the Monatshefte Fiir Mathematik und Physik, Volume 38, pages 173- :198 (Leipzig, 1931). Godel intended to write a second part to his paper, but illness prevented him from publishing it. Godel's theorem is sometimes regarded as the most decisive result 1 in mathematical logic. In the paper which follows we shall familiarize ;ourselves with the mathematics (metamathematics) required to understand •this theorem. iii TABLE OF CONTENTS Page(s) PREFACE iii TABLE OF CONTENTS iv-v ABSTRACT vi-vii I. INTRODUCTION l A. THE STATE OF MATHEMATICS @ 1930 l B. THE CREATION OF METAMATHEMATICS - 3 PROOF THEORY II. SUPPORTING DETAILS FOR THE PROOF 8 A. GODEL NUMBERING - ARITHMETIZATION OF 8 METAMATH11MATICS B. SELF-REFERENTIAL STATEMENTS 15 1. Formulation of the Richard Paradox 15 2. The Richard Paradox 16 3. Resolving the Paradox 17 C. THE FORMAL SYSTEM 19 1. Basic Symbols 20 2. Well-Formed Formulas (WE'F' s) 21 3· Axioms (Initial Formulae) 23 4. "Immediate Consequence of" 24 D. PRIMITIVE RECURSIVE FUNCTIONS 24 1. Significance 24 2. Definitions 25 3. Definition by Induction - Recursive 27 Definition 4. Primitive Recursive 28 iv TABLE OF CONTENTS ~age(s) 5. Examples of Primitive Recursive Functions 31 6. Historical Utilization 33 III. GODEL' s PROOF 34 A. STATEMENT OF THE THEOruMS 34 B. TARSKI' S EXAMPLE :;4 C. GOOEL 1 S ASSUMPTIONS 35 D. LEMMA 1 38 E. DEFINITION: ¢ (x,y) 38 F. EXPRESSIBILITY 39 G. "Bew(x)" 39 H. LEMMA 2 4o I. UMMA 3 4o J. GODEL Is SECOND THEOREM 41 K. CONCLUSIONS 41 IV. EPILOGUE - TRUTH IN MATHEMATICS 43 ANNOTATED BIBLIOGRAPHY 45 INDEX OF DEFINITIONS 48 Photograph of Kurt GOdel by Orren J. Turner viii •rables: Table 1 - Godel Numbers of Constant Signs 9 Table 2 - Godel Numbers of Type 1 Variables 9 Table 3 - Godel Numbers of Type 2 Variables 10 Table 4 - Godel Numbers of Type 3 Variables 10 v ABSTRACT GODEL'S 11 0N FORMALLY UNDECIDABLE PROPOSI'l'IOOS ••• 11 by Mary Margaret Pepe Master of Sciences in Mathematics, Option II Godel's theorem is composed of two parts: Theorem 1: For suitable formal systems L, there are undecidable pro+ I positions in L; that is propositions F such that neither F nor not F is provable. Theorem 2: For suitable L, the simple consistency of L cannot be proved in L. Part One of the thesis examines the state of mathematics @ 1930 and !the creation of proof theory - metamathematics. I I Part Two presents the supporting details for the proot including: l) GOdel numbering - the establishing of a one-to-one correspon~ dence between the finite expressions of the system and the natural numbers - is presented. I 2) Belt-referential statements - statements which talk about them The Richard Paradox is formulated, I selves - are described. l vi examined, and resolved. -----· ··-· --~ 3) The Formal System L is described - its basic signs, well-formet· formulas, axioms, and the relation "Immediate consequence of." 4) Primitive Recursive Functions are defined, examples shown, and historical utilization described. Part Three presents a Rosser version of the proof of theorems 1 and 2 stated above. Reference is made to primitive recursive functions and numeralwise expressibility. Part Four, an epilogue, considers ramifications regarding truth in mathematics. An index of definitions and an annotated bibliography follow the thesis. ____ _jI vii I. INTRODUCTION 1 I A. THE STATE OF MATHEMATICS @ 19.30 At the second mathematical congress held in Paris in 1900, David I ! Hilbert posed twenty-three problems which he thought would be among I those occupying the attention of the twentieth-century mathematicians. I ! This paper is in partan answer to Hilbert's second query as to whether: the axioms of arithmetic are consistent. I I Before we continue we need to have a common understanding of what i is meant by consistent, complete, incomplete, decidable, undecidable, I provable, and disprovable. We will give naive definitiorJs and when I I necessary make these definitions more explicit when we an9.lyze the I metamathematics of the formal system. I Con:::t:::tem of axioms is said to be consistent if a finite number of I logical steps based on the axioms would never lead to a contradiction. ~.~! I Inconsistent Not consistent. I Complete I in :~:·:::t:: :~: ::g::.:o::::::u::c:v::yt::-:.:::::ment expressible I Incomplete j A system is said to be incomplete if ?£t eve!Y tru~ statement of the! system is deducible from the axioms. Decidable A formula P written in the vocabulary of a given set of axioms is (1) (2) said to be decidable if there is an effective method (see page 5) for determining whether P or not P is deducible from the axioms. Undecidable A formula P is said to be undecidable if neither P nor not P is a theorem. Provable A formula P is provable, i.e., is a theorelii, iff there is a formal proof of the formula ending in P. A formal proof of a formula P is a finite column of formulas each of whose lines is an axiom or may be inferred from the preceding lines by specified rules of inference (such as modus ponE:ms : A, A--") B, ;. B). Unprovable Cannot be proved. Bertrand Russell and Alfred North Whitehead (1861-1947) had made an elaborate e.ttempt to develop the fundame.ntal notions of arithmetic / from a precise set of axioms. Their text was in the tradition of Leibniz, Boole, and Frege and was based on Peano's axioms. Russell and Whitehead intended to prove that all of pure mathematics could be derived from a small number of fundamen-tal logical principles. If they had succeeded their proof would have supported Russell's view that mathematics is indistinguishable from logic. In 1926 Finsler published his paper "Formal proofs and undecid- ability." Finsler presented a proposition that although false is formally undecidable. In 1930 Presburger published a decision pro- cedure applicable to every proposition of a system of arithmetic which used only addition, but not multiplication. Presburger proved that every one of its propositions is decidable - either provable or unprovable. Hilbert had hoped to prove the consistency of elementary number theory with the possibility of reducing other portions of classical mathematics to that of the natural numbers via models. Work along these lines carne to somewhat of a halt in 1931 when GOdel demonstrated the impossibility of proving the consistency of I any formal theory by constructive methods formalizable within the theory itself. This includes the formulas of the Natural numbers. Godel proved that if the fot'mal .system is consistent then it is incomplete. Nineteenth-century mathematicians had hoped that the axiomc; for ari thmetio· were complete or could be made complete by the .. addition of a finite number of further axioms. ~1e discovery that this.·. was not so was more of a shock than Godel 1 s demonstrating the existence of undecidable propositions. B. THE CREATION OF METAMATHEMATICS - PROOF THEORY Metamathematics, which is the study of rigorous proof in mathe­ matics, was developed by Hilbert during the 1920's-1930's. Metamath­ ematics looks at mathematics from the outside. It is concern~;d with the interpretations of signs and rules and not with the operatlons of arithmetic. Godel's theorem is a result which belongs to metamathematics. Godel put the notion of a formal system at the very center of his investigations~ We thus need to define the following formal notions: (4) Formal When we use the word formal we are referring only to the forms of certain expressions and not to their meaning. •Formal Theory A formal theory is a completely symbolic language built according 'to certain rules from an alphabet of specified primitive symbols. String A finite sequence of formal symbols. Object Language The symbolic ( f\)nnal) language in which the statements of the formal! i :theory are written. A statement in the object language is a statement !.2! the theory. I I I iHetalanguage - Syntax Language i I The language used to present the formal theory. A statement in i the metalanguage is a statement about the theory. It is used to discus$ i the fonnal theory, which includes defining its syntax, specifying its .axioms and rules of inference, and analyzing its properties. Meta theorem A theorem about a formal theory (written in English). A theorem of the theory is written jt1 the symbolism of the theory. A metatheorem 'is a statement that is proved in the metalanguage (English) about the· theory. 'Metatheorz Metatheory is the theory of formal languages and systems and their interpretations. It takes formal languages and systems and their in- :terpretations as objects of study, and consists in the body of truths (5) and conjectures about these objects. Among its main problems are problems about the consistency, completeness, decidability and i!!,­ _?ependence of sets of formulas.