A-Friendly-Introduction-to-Mathematical-Logic

A Fr i en o troduction n I y t dl Mathematical Logic 2nd Edition Christopher C Leary Lars Kristiansen A Friendly Introduction to Mathematical Logic A Friendly Introduction to Mathematical Logic 2nd Edition Christopher C Leary State University of New York College at Geneseo Lars Kristiansen The University of Oslo Milne Library, SUNY Geneseo, Geneseo, NY c 2015 Christopher C Leary and Lars Kristiansen ISBN: 978-1-942341-07-9 (paperback) ISBN: 978-1-942341-32-1 (ebook) This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License You are free to: Share—copy and redistribute the material in any medium or format Adapt—remix, transform, and build upon the material The licensor cannot revoke these freedoms as long as you follow the license terms Under the following terms: Attribution—You must give appropriate credit, provide a link to the license, and indicate if changes were made You may so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use NonCommercial—You may not use the material for commercial purposes ShareAlike—If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original Milne Library SUNY Geneseo One College Circle Geneseo, NY 14454 Lars Kristiansen has received financial support from the Norwegian Non-fiction Literature Fund Contents Preface ix Structures and Languages 1.1 Naăvely 1.2 Languages 1.2.1 Exercises 1.3 Terms and Formulas 1.3.1 Exercises 1.4 Induction 1.4.1 Exercises 1.5 Sentences 1.5.1 Exercises 1.6 Structures 1.6.1 Exercises 1.7 Truth in a Structure 1.7.1 Exercises 1.8 Substitutions and Substitutability 1.8.1 Exercises 1.9 Logical Implication 1.9.1 Exercises 1.10 Summing Up, Looking Ahead 12 13 17 19 21 22 26 27 32 33 36 36 38 38 Deductions 2.1 Naăvely 2.2 Deductions 2.2.1 Exercises 2.3 The Logical Axioms 2.3.1 Equality Axioms 2.3.2 Quantifier Axioms 2.3.3 Recap 2.4 Rules of Inference 2.4.1 Propositional Consequence 2.4.2 Quantifier Rules 41 41 43 47 48 48 49 49 50 50 53 v vi CONTENTS Completeness and Compactness 3.1 Naăvely 3.2 Completeness 3.2.1 Exercises 3.3 Compactness 3.3.1 Exercises 3.4 Substructures and the Lă owenheimSkolem 3.4.1 Exercises 3.5 Summing Up, Looking Ahead Theorems 73 73 74 86 87 93 94 101 102 Incompleteness from Two Points of View 4.1 Introduction 4.2 Complexity of Formulas 4.2.1 Exercises 4.3 The Roadmap to Incompleteness 4.4 An Alternate Route 4.5 How to Code a Sequence of Numbers 4.5.1 Exercises 4.6 An Old Friend 4.7 Summing Up, Looking Ahead 2.5 2.6 2.7 2.8 2.9 2.4.3 Exercises Soundness 2.5.1 Exercises Two Technical Lemmas Properties of Our Deductive System 2.7.1 Exercises Nonlogical Axioms 2.8.1 Exercises Summing Up, Looking Ahead 54 54 58 58 62 65 66 70 71 103 103 105 107 108 109 109 112 113 115 Syntactic Incompleteness—Groundwork 5.1 Introduction 5.2 The Language, the Structure, and the Axioms of N 5.2.1 Exercises 5.3 Representable Sets and Functions 5.3.1 Exercises 5.4 Representable Functions and Computer Programs 5.4.1 Exercises 5.5 CodingNaăvely 5.5.1 Exercises 5.6 Coding Is Representable 5.6.1 Exercise 5.7 Gă odel Numbering 5.7.1 Exercises 117 117 118 119 119 128 129 133 133 136 136 139 139 142 CONTENTS 5.8 5.9 5.10 5.11 5.12 5.13 Gă odel Numbers and N 5.8.1 Exercises Num and Sub Are Representable 5.9.1 Exercises Definitions by Recursion Are Representable 5.10.1 Exercises The Collection of Axioms Is Representable 5.11.1 Exercise Coding Deductions 5.12.1 Exercises Summing Up, Looking Ahead vii The Incompleteness Theorems 6.1 Introduction 6.2 The Self-Reference Lemma 6.2.1 Exercises 6.3 The First Incompleteness Theorem 6.3.1 Exercises 6.4 Extensions and Refinements of Incompleteness 6.4.1 Exercises 6.5 Another Proof of Incompleteness 6.5.1 Exercises 6.6 Peano Arithmetic and the Second Incompleteness 6.6.1 Exercises 6.7 Summing Up, Looking Ahead Computability Theory 7.1 The Origin of Computability Theory 7.2 The Basics 7.3 Primitive Recursion 7.3.1 Exercises 7.4 Computable Functions and Computable Indices 7.4.1 Exercises 7.5 The Proof of Kleene’s Normal Form Theorem 7.5.1 Exercises 7.6 Semi-Computable and Computably Enumerable 7.6.1 Exercises 7.7 Applications to First-Order Logic 7.7.1 The Entscheidungsproblem 7.7.2 Gă odels First Incompleteness Theorem 7.7.3 Exercises 7.8 More on Undecidability 7.8.1 Exercises 142 147 147 153 153 156 156 158 158 166 167 Theorem 169 169 170 173 174 181 182 185 185 187 187 192 193 Sets 195 195 197 204 212 215 223 225 233 235 242 244 244 248 253 254 262 viii Summing Up, Looking Ahead 8.1 Once More, With Feeling 8.2 The Language LBT and the Structure B 8.3 Nonstandard LBT -structures 8.4 The Axioms of B 8.5 B extended with an induction scheme 8.6 Incompleteness 8.7 Off You Go CONTENTS 265 266 266 271 271 274 276 278 Appendix: Just Enough Set Theory to Be Dangerous 279 Solutions to Selected Exercises 283 Bibliography 359 Preface Preface to the First Edition This book covers the central topics of first-order mathematical logic in a way that can reasonably be completed in a single semester From the core ideas of languages, structures, and deductions we move on to prove the Soundness and Completeness Theorems, the Compactness Theorem, and Găodels First and Second Incompleteness Theorems There is an introduction to some topics in model theory along the way, but I have tried to keep the text tightly focused One choice that I have made in my presentation has been to start right in on the predicate logic, without discussing propositional logic first I present the material in this way as I believe that it frees up time later in the course to be spent on more abstract and difficult topics It has been my experience in teaching from preliminary versions of this book that students have responded well to this choice Students have seen truth tables before, and what is lost in not seeing a discussion of the completeness of the propositional logic is more than compensated for in the extra time for Gă odel’s Theorem I believe that most of the topics I cover really deserve to be in a first course in mathematical logic Some will question my inclusion of the Lă owenheim–Skolem Theorems, and I freely admit that they are included mostly because I think they are so neat If time presses you, that section might be omitted You may also want to soft-pedal some of the more technical results in Chapter The list of topics that I have slighted or omitted from the book is depressingly large I not say enough about recursion theory or model theory I say nothing about linear logic or modal logic or second-order logic All of these topics are interesting and important, but I believe that they are best left to other courses One semester is, I believe, enough time to cover the material outlined in this book relatively thoroughly and at a reasonable pace for the student Thanks for choosing my book I would love to hear how it works for you ix 351 By B6, we have B a1 = a2 → (0 ◦ a1 = ◦ a2 ∧ ◦ a1 = ◦ a2 ) By (1), (2) and (PC), we have B (2) b1 = b2 Case (ii): In this case we have B b1 = b2 straightaway by B5 We not need the induction hypothesis Exercise 8.4: Assume B |= t1 = t2 (we will prove B t1 = t2 ) There exists one, and only, one biteral b1 such that B |= t1 = b1 , and there exists one, and only one, biteral b2 such that B |= t2 = b2 By Exercise 8.4 and the Soundness Theorem, we have B t1 = b1 and B t2 = b2 Now, B |= b1 = b2 (this holds since B |= t1 = t2 ) By Exercise 8.4, we have B b1 = b2 By using logical axioms, we can deduce the formula t1 = t2 from the formulas t1 = b1 and t2 = b2 and b1 = b2 Hence B t1 = t2 Exercise 8.4: Let φ(x1 , xn ) be an existential LBT -formula where all the free variables are displayed For any variable free LBT -terms t1 , tn , we have B |= φ(t1 , tn ) ⇒ B φ(t1 , tn ) (*) We prove (*) by induction on the structure of φ Our proof contains one case for each clause in the definition of an existential formula Case (i): φ is an atomic formula In this case (*) holds by Exercise 8.4 Case (ii): φ is of the form ¬ψ where ψ is an atomic formula In this case (*) holds by Exercise 8.4 Case (iii): φ is of the form (α ∧ β) Assume B |= (α ∧ β) (We will prove B (α ∧ β) ) Then, as we know that B |= α and B |= β, our induction hypothesis yields B α and B β By (PC), we have B (α ∧ β) Case (iv): φ is of the form (α ∨ β) This case is similar to (iii) Case (v): φ is of the form (∃x)(ψ) Now, ψ will be of the form ψ(x, t1 , , tm ) where t1 , , tm are variable free LBT -terms Assume B |= (∃x)ψ(x, t1 , , tm ) (We will prove B (∃x)ψ(x, t1 , , tm ) ) 352 Solutions to Selected Exercises There exists b ∈ {0, 1} such that B |= ψ(b, t1 , , tm ) (Such a b B exists since B |= (∃x)ψ(x, t1 , , tm ) Recall that b = b and that b is a variable free LBT -term.) By our induction hypothesis, we have B ψ(b, t1 , , tm ) (1) and ψ(b, t1 , , tm ) → (∃x)ψ(x, t1 , , tm ) (Q2) is one of the logical axioms of our proof calculus By (1), (Q2) and (PC), we have B (∃x)ψ(x, t1 , , tm ) It follows straightforwardly from (*) that any existential sentence true in B can be deduced from the axioms of B (There are no free variables in an existential sentence.) Section 8.5, page 274 Exercise 8.5: This follows from the Soundness Theorem This is one way to put it: The Soundness Theorem states that any formula deducible from B is true in all models for B If we can find a structure A such A |= B and A |= φ, we know that φ is not true in all models of B By the Soundness Theorem, we can conclude that φ is not deducible from B This is another way to put it: The notation B |= φ means that φ is true in all models for B The Soundness Theorem states that B φ ⇒ B |= φ (*) Assume A |= B and A |= φ Then, we have B |= φ By B |= φ and (*), we have B φ Exercise 8.5: We will build an LBT -structure A such that A |= B and A |= (∀x) x = e → (∃y)[0 ◦ y = x ∨ ◦ y = x] The universe of A is {a, 0, 1} , that is, the set of all finite strings over the ternary alphabet {a, 0, 1} Furthermore, eA = ε and 0A = and 1A = and ◦A is the standard concatenation of two strings Now, consider any string in the universe that starts with the symbol a, e.g., the string a101 This string is not the result of concatenating with another string in the universe: We have a101 = ◦A t for any t ∈ {a, 0, 1} Neither, is the string the result of concatenating 353 with another string in the universe Moreover, the string is not the empty string Hence, we have A |= x = e → (∃y)[0 ◦ y = x ∨ ◦ y = x] s[x|a101] for some assignment function s Hence A |= (∀x) x = e → (∃y)[0 ◦ y = x ∨ ◦ y = x] It is easy to see that all the axioms of B are true in A Exercise 8.5: It follows from the Soundness Theorem that both B1 and B2 are consistent The Soundness Theorem states that Σ φ ⇒ Σ |= φ This is equivalent to Σ has a model ⇒ Σ is consistent Now, B is a model for B1 Thus, B1 is consistent In the previous exercise we constructed a model A for B2 Thus, B2 is consistent Exercise 8.5: We need infinite bit strings Let ω denote the length of a bit string that contains an ith bit for each i ∈ N (The readers familiar with ordinal numbers will know why we have picked the Greek letter ω to denote this length.) A bit string of length ω starts somewhere, that is, there is a first element, a second element, and so on, but it does not end somewhere, that is, there is no last element Infinite bit strings can be concatenated with finite and infinite bit strings Let us study some examples Let α be the bit string of length ω that contains nothing but zeros Then, 101α is the string where the 1st bit is 1; where the 2nd bit is 0; where the 3rd bit is and, for any i > 3, the ith bit of 101α is So if we put the finite bit string 101 in front of the infinite bit string α, we get a bit string of length ω If we put the infinite bit string α in front of the finite bit string 101, we get a bit string that is longer than ω For all i > 0, the ith bit of α101 is In position ω + of α101, we find the bit 1; in position ω + we find 0; and in position ω + we find Then the string stops The string α101 is of length ω + Let α be the bit string of length ω that contains nothing but zeros, and let β be the bit string of length ω that contains nothing but ones The string βα is a string of length ω + ω So is the string αα The string βα consists of an infinite sequence of ones followed by an 354 Solutions to Selected Exercises infinite sequence of zeros, whereas the string αα consists of an infinite sequence of zeros followed by another infinite sequence of zeros What is the length of the string α101α? Well, the string is of this form 0000000000 1010000000 ω bits ω bits so the string is of length ω + ω The length of the string αα10101 is ω + ω + since the string is of the form 000000000 000000000 ω bits ω bits 10101 bits n copies of ω Let ωn denote ω + + ω For any n, m ∈ N, it makes sense to talk about bit strings of length ωn + m A string of length ωn + m will be of the form n strings of length ω ω bits ω bits m bits Let A be a LBT -structure where the universe A is {α | ∃m, n ∈ N and α is a bit string of length ωn + m} Furthermore, let 0A = and 1A = and eA = ε (ε is the only string of length 0) Finally, let ◦A be concatenation (see the examples above) Then, A |= B Furthermore, when α is, e.g., the bit string of length ω that contains only zeros, we have 0◦A α = α Thus, A |= (∀x)0◦x = x Exercise 8.5: Let φ(x) :≡ ◦ x = x We will prove that BI (∀x)φ(x) By B4, we have BI φ(e) (1) Next we need the axiom (∀x)(∀y)[ x = y → (0 ◦ x = ◦ y ∧ ◦ x = ◦ y) ] (B6) By this axiom, we have (use ◦ x for x, and use x for y) BI ◦ x = x → [0 ◦ (0 ◦ x) = ◦ x ∧ ◦ (0 ◦ x) = ◦ x] (2) By (2) and (PC), we have BI ◦ x = x → ◦ (0 ◦ x) = ◦ x (3) 355 Now we need the axiom (∀x)(∀y)0 ◦ x = ◦ y (B5) By this axiom, we have (use ◦ x for x and use x for y) BI ◦ (1 ◦ x) = ◦ x (4) By (3), (4), and (PC), we have BI ◦ x = x → [0 ◦ (0 ◦ x) = ◦ x ∧ ◦ (1 ◦ x) = ◦ x] (5) The formula in (5) is of the form φ(x) → (φ(0 ◦ x) ∧ φ(1 ◦ x)) Thus, BI (∀x)[φ(x) → (φ(0 ◦ x) ∧ φ(1 ◦ x))] By (1), (6), the scheme (I), and (PC), we have BI (6) (∀x)φ(x) Section 8.6, page 276 Exercise 8.6: We work through the various clauses of the inductive proof: First, assume that f is the successor function S Let φ(x, y) :≡ 1◦x = y Assume f is the project function Iin Let φ(x1 , , xn , y) :≡ x1 = x1 ∧ ∧ xn = xn ∧ xi = y If f is the zero function O, let φ(y) :≡ y = For the composition case, assume that f (x) = h(g1 (x), , gm (x)) ∼ ∼ ∼ where x = x , , x To improve the readability, we will assume n ∼ that n = By our induction hypothesis we have LBT -formulas ψ1 , , ψm and ξ such that gi (a) = b ⇔ B |= ψi (1a , b) (for i = 1, , m) and h(a1 , , am ) = b ⇔ B |= ξ(1a1 , , 1am , 1b ) Let φ(x, y) be the formula (∃z1 ) (∃zm )[ψ1 (x, z1 ) ∧ ∧ ψi (x, zm ) ∧ ξ(z1 , , zm , y)] Assume f (x, 0) = g(x) and ∼ ∼ f (x, z + 1) = h(x, z, f (x, z)) ∼ ∼ ∼ 356 Solutions to Selected Exercises where x = x1 , , xn To improve the readability, we will assume ∼ that n = The induction hypothesis yields LBT -formulas ψ(x, y) and ξ(x, z1 , z2 , y) such that g(a) = b ⇔ B |= ψ(1a , 1b ) and h(a, c1 , c2 ) = b ⇔ B |= ξ(1a , 1c1 , 1c2 , 1b ) Recall the formula slessthan(x, y) from Exercise 8.2 Let φ(x, z, y) be the formula (∃t)(∃u0 ) IthElement(u0 , 1, t) ∧ ψ(x, u0 ) ∧ IthElement(y, z◦1, t) ∧ (∀i) slessthan(i, z) → (∃u)(∃v)[IthElement(u, i ◦ 1, t) ∧ IthElement(v, i ◦ ◦ 1, t) ∧ ξ(x, i ◦ 1, u, v)] Assume f (x) = (µi)[g(x, i)] where x = x1 , , xn To improve the ∼ ∼ ∼ readability, we will assume that n = By the induction hypothesis, we have an LBT -formula ψ(x, i, y) such that g(a, i) = b ⇔ B |= ψ(1a , 1i , 1b ) Let φ(x, y) be the formula (∀i) slessthan(i, y) → (∃u)[ψ(x, i, u) ∧ ¬(u = e)] ∧ ψ(x, y, e) (Recall that e = 10 ) Exercise 8.6: The set K is semi-computable Thus, we have a computable function f such that dom(f ) = K Let g(x) = − f (x) Then, g is a computable function such that g(a) = iff a ∈ K By Exercise 8.6 we have an LBT formula ψ(x, y) such that B |= ψ(1a , 1b ) iff g(a) = b Let φ(x) :≡ ψ(x, 10 ) Then, we have B |= φ(1a ) iff a ∈ K Exercise 8.6: Note that e = 29 and that we have ◦1t = 215 313 t for any LBT -term t Furthermore, note that 10 = e and that 1a+1 = ◦11a Let g(0) = 29 and g(y + 1) = 215 313 5g(y) Then, we have g(a) = 1a The function g is defined from primitive recursive functions by composition and primitive recursion Thus, g is a primitive recursive function Next we define the function ft by induction on the structure of the LBT -term t Let • ft (a) = g(a) if t is the variable x 357 • ft (a) = 22i if t is the variable vi and vi is different from x • ft (a) = 29 if t is e • ft (a) = 211 if t is • ft (a) = 213 if t is • ft (a) = 215 3ft1 (a) 5ft2 (a) if t is ◦t1 t2 We have ft (a) = tx1a where tx1a denotes the term t where each occurrence of the variable x is replaced by the term 1a For each t, the function ft is defined by composition of primitive recursive functions Hence, for each (fixed) term t, the function ft is primitive recursive Finally, we define the function fφ by induction on the structure of the formula φ Let • fφ (a) = 21 3fα (a) if φ is (ơ) ã f (a) = 23 3f (a) 5f (a) if φ is (α ∨ β) • fφ (a) = 25 3fvi (a) 5fα (a) if φ is (∀vi )(α) • fφ (a) = 27 3ft1 (a) 5ft2 (a) if φ is = t1 t2 Now, we have fφ (a) = φ(1a ) Moreover, for each fixed formula φ, the function fφ is defined by composition of primitive recursive functions Thus, fφ is primitive recursive Exercise 8.6: Recall that a set is semi-computable iff it is the domain of a computable function (this is the definition of a computable set) Since { η | A η} is semi-computable, we have a computable function g such that dom(g) = { η | A η} Let fφ be the primitive recursive function from Exercise 8.6 Then, fφ (a) = φ(1a ) Let h(x) = g(fφ (x)) Now, h is a computable function since g and fφ are computable functions Moreover, dom(h) = {a | A φ(1a )} Hence, {a | A φ(1a )} is a semi-computable set Exercise 8.6: By Exercise 8.6, we have a formula φ(x) such that B |= φ(1a ) iff a ∈ K Thus {a | B |= ¬φ(1a )} (1) is the set K, and we know that K is not a semi-computable set By Exercise 8.6, we know that {a | A ¬φ(1a )} (2) 358 Solutions to Selected Exercises is a semi-computable set This means that (1) and (2) cannot be the same set as latter is a semi-computable set whereas the former is not We have assumed that B |= A Thus, we have A ¬φ(1a ) ⇒ B |= ¬φ(1a ) by the Soundness Theorem From this we can conclude that (2) has to be a strict subset of (1) Thus, there must be a natural number a such that B |= ¬φ(1a ) and A ¬φ(1a ) Exercise 8.6: It is easy to see that the set { η | BI η} semi-computable An algorithm can verify if a formula η is deducible from the axioms of BI The algorithm can, e.g., generate all possible BI -deductions one by one If the algorithm encounters a deduction of η, it halts If the algorithm never encounters such a deduction, it runs forever Thus, { η | BI η} is a semi-computable set and, by Exercise 8.6, we have θ such B |= θ and BI θ Exercise 8.6: By the previous exercise we have θ such B |= θ and BI θ Moreover, we have B |= BI and B |= ¬θ Thus, by the Soundness Theorem, we have BI ¬θ Bibliography [Barwise 77] Jon Barwise, ed Handbook of Mathematical Logic Amsterdam: North-Holland, 1977 [Bell and Machover 77] John L Bell and Mosh´e Machover A Course in Mathematical Logic Amsterdam: North-Holland, 1977 [Boolos 89] George Boolos A New Proof of the Găodel Incompleteness Theorem Notices of the American Mathematical Society, Vol 36, No 4, April 1989, pp 388390 Gă odel’s Second Incompleteness Theorem Explained [Boolos 94] in Words of One Syllable Mind, Vol 103, January 1994, pp 1–3 [Chang and Keisler 73] C C Chang and H J Keisler Model Theory Amsterdam: North-Holland, 1973 [Crossley et al 72] J N Crossley, C J Ash, C J Brickhill, J C Stillwell, and N H Williams What Is Mathematical Logic? London: Oxford University Press, 1972 [Dawson 97] John W Dawson, Jr Logical Dilemmas: The Life and Work of Kurt Gă odel Wellesley, Mass.: A K Peters, 1997 [Enderton 72] Herbert B Enderton A Mathematical Introduction to Logic Orlando, Fla.: Academic Press, 1972 [Feferman 60] Solomon Feferman Arithmetization of Metamathematics in a General Setting Fundamenta Mathematicae, Vol 49, 1960, pp 35–92 In the Light of Logic Oxford: Oxford University [Feferman 98] Press, 1998 [Gă odelWorks] Kurt Gă odel Collected Works: Vol I, Publications 1929– 1936 Edited by Soloman Feferman, John W Dawson, Jr., Stephen C Kleene, Gregory H Moore, Robert M Solovay, and Jean Van Heijenoort New York: Oxford University Press, 1986 359 360 BIBLIOGRAPHY [Goldstern and Judah 95] Martin Goldstern and Haim Judah.The Incompleteness Phenomenon: A New Course in Mathematical Logic Wellesley, Mass.: A K Peters, 1995 [Henle 86] James M Henle An Outline of Set Theory New York: SpringerVerlag, 1986 [Hofstadter 85] Douglas R Hofstadter Metamagical Themas: Questing For The Essence Of Mind And Pattern Basic Books, 1985 [Hrbacek and Jech 84] Karel Hrbacek and Thomas Jech Introduction to Set Theory New York: Marcel Dekker, 1984 [Keisler 76] H Jerome Keisler Foundations of Infinitesimal Calculus Boston: Prindle, Weber & Schmidt, 1976 [Keisler and Robbin 96] H Jerome Keisler and Joel Robbin Mathematical Logic and Computability New York: McGraw-Hill, 1996 [Malitz 79] Jerome Malitz Introduction to Mathematical Logic New York: Springer-Verlag, 1979 [Manin 77] Yu I Manin A Course in Mathematical Logic New York: Springer-Verlag, 1977 [Mendelson 87] Elliott Mendelson Introduction to Mathematical Logic Monterey, Calif.: Brooks/Cole, a division of Wadsworth, 1987 [Roitman 90] Judith Roitman Introduction to Modern Set Theory New York: Wiley, 1990 [Russell 67] Bertrand Russell The Autobiography of Bertrand Russell, Vol I London: George Allen & Unwin, 1967 [Turing 37] A.M Turing On Computable Numbers, with an Application to the Entscheidungsproblem Proceedings of the London Mathematical Society Vol s2–42, No 1, 1937, pp 230–265 Index Ackermann function, 214 alphabet, 266 arity, assignment function term, 28 variable, 28 x-modification, 28 atomic formula, 11 axiom equality, 48–49 logical, 48–50 quantifier, 49 axiomatized, 174 completeness Σ-, 114 Completeness Theorem, 75 composition, 198 computable function, 130, 202 computable index, 215 computable set, 203 semi-, 238 computably enumerable set, 238 computation, 226 consistent, 74 countable, 97, 280 decidable, 48 deduction, 43 Deduction Theorem, 64 definable set, 123 defines, 123 ∆-formula, 107 derivability conditions for Peano Arithmetic, 188 Diophantine equation, 255 double recursion, 215 dwarfs, seven, 279 Berry’s paradox, 185 bit, 266 biteral, 268 Boolos, George, 185 Brouwer, L.E.J., calculable function, 131 Cantor’s Theorem, 280 Cantor, Georg, 280 cardinality, 279 cases definition by, 207 characteristic function, 128, 203 Church’s Thesis, 196 Church, Alonzo, 130, 195 Church’s Thesis, 131, 132 Compactness Theorem, 87 complete deductive system, 74 set of axioms, 104 complete diagram, 102 elementarily equivalent, 89 elementary chain, 101 elementary extension, 95 elementary substructure, 95 Entscheidungsproblem, 195, 221, 245 Enumeration Theorem, 219 equality axiom, 48–49 existential formula, 273 extension by constants, 77 361 362 INDEX Feferman, Solomon, 191 finitely satisfiable set of formulas, 87 formula, 11 atomic, 11 ∆, 107 Π, 106 Σ, 106 Frege, Gottlob, function Ackermann, 214 calculable, 131 characteristic, 128, 203 computable, 130, 202 partial, 121 primitive recursive, 203 projection, 198 recursive, 130 representable, 121 total, 121 weakly representable, 121 well-defined, 81–82 function composition, 198 isomorphic, 27 isomorphism, 27 Gă odel, Kurt, 2, 75, 130 Gă odel number, 139 m-reducible, 244 minimalization bounded, 208 minimalization operator, 201 model, 31 model of arithmetic, 89 nonstandard, 88 modus ponens, 45 MRDP Theorem, 257 µ-operator, 201 Halting Problem, 221 Undecidability of the, 221 Harrington, Leo, 191 Henkin axiom, 78 Henkin constant, 78 Henkin, Leon, 26, 75 Hilbert’s 10th Problem, 256 Unsolvability of, 257 Hilbert, David, 1, 188 Hobbes, Thomas, 41 Incompleteness Theorem First, 176, 250 Second, 189 inconsistent, 74 index computable, 215 induction on complexity, 15 initial segment, 18 Kleene T -predicate, 218, 233 Kleene’s Normal Form Theorem, 217 Kleene, Stephen, 197 Kă onigs Infinity Lemma, 92 L-structure, 23 λ-calculus, 130 language, least number operator, 201 length, 111 liar paradox, 185 linear order, 94 listable set, 133 Lă obs Theorem, 192 logical axiom, 4850 logical implication, 36 Lă owenheim–Skolem Theorem Downward, 97 Upward, 100 N , 67 n-ary, names, 186 nonstandard analysis, 89 Normal Form Theorem, 217 notation infix, 10 Polish, 10 number theory language of, 19 numeralwise determined, 129 INDEX ω-consistent, 182 Padding Lemma, 216, 223 Paris, Jeff, 191 partial function, 121 Peano Arithmetic first-order, 188 second-order, 100 Π-formula, 106 positively numeralwise determined, 129 Presburger Arithmetic, 254 prime component of a formula, 161 primitive recursion, 199 primitive recursive function, 203 primitive recursive set, 203 Principia Mathematica, 249 projection function, 198 propositional consequence in first order logic, 52 in propositional logic, 52 provable formula, 126 quantifier bounded, 105 scope of, 11 quantifier axiom, 49 R, 89 recursion definition by, 10 double, 215 primitive, 199 recursive function, 130 recursively axiomatized, 174 refutable formula, 126 represent, 120 representable function, 121 representable set, 120 restriction of a function, 95 of a model, 84 Rice’s Theorem, 259 Robinson, Abraham, 89 Rosser’s Lemma, 125 Rosser’s Theorem, 183 363 Rosser, John Barkley, 183, 250 rule of inference type (PC), 53 type (QR), 53 Russell, Bertrand, 44, 249 S-m-n Theorem, 222 satisfaction, 29 satisfiable set of formulas, 87 Schră oderBernstein Theorem, 280 scope, 11 Self-Reference Lemma, 172 semi-computable set, 238 sentence, 20 sequence code, 159 set computable, 203 primitive recursive, 203 Σ-completeness, 114 Σ-formula, 106 Skolem function, 99 Skolem’s paradox, 99 Soundness Theorem, 57 string, 266 structure, 23 Henkin, 26 universe of, 23 subformula, 11 substitutable, 35 substructure, 94 elementary, 95 Tarski’s Theorem, 180 tautology of propositional logic, 50–51 term, term assignment function, 28 term construction sequence, 144 theory, 174 of a set of formulas, 174 of a structure, 89, 174 theory of N, 104 ThmΣ , 46 total function, 121 tree, 92 364 true, 31 Turing machine, 130, 221 Turing’s Thesis, 196 Turing, Alan, 130, 195, 221 uncountable, 97 unique readability, 10 universal closure, 38 Universal Function Theorem, 220 universe, 23 size of the, 141 valid, 37 variable bound, 21 free, 20 variable assignment function, 28 weakly represent, 120 weakly representable function, 121 weakly representable set, 120 well-defined function, 81–82 well-order, 94 Whitehead, Alfred North, 249 INDEX List of Symbols φxt , 34 A, 23 A |= φ[s], 29 A |= φ, 31 A ⊆ B, 94 A ≺ B, 95 · , 110 A , 266 rng, 238 S, 19 s[x|a], 28 Σ φ, 43 S, 198 :≡, βP , 51 BI , 275 Tn , 218, 233 T h(A), 174 T h(A), 89, 174 ThmΣ , 46 χA , 203 , 226 , 112 Con PA , 189 U (t), 233 U(t), 218 uxt , 34 (·)i , 111 ∆ |= Γ, 36 dom, 238 Vars, E, 19 {e}n , 218 We , 241 Iin , 198 x, 121 ∼ x, 121 ∼ K, 242 O, 198 ≡, 89 ⊥, 74 φ , 139 ∼ =, 27 |= φ, 37 , 84, 95 L, LN T , 19 LR , 89 ≤m , 244 N , 67 N, 23 N