INDEX

Abel, Niels, 342 scheme, 228 Abelian group, 342 Axiom(s), 24 Absolute value, 116 choice, 231, 235 Abstraction, 122 empty , 227 Addition, rule, 25 equality, 226 Addition, matrix, 355 , 227 Additive identity, 199, 412 foundation, 234 Additive inverse, 199 Frege–Łukasiewicz, 24 Aleph, 313 group, 340 Algebraic, 315 paring, 228 Alphabet, 5 , 228 first-order, 68 regularity, 234 second-order, 73 replacement, 230 And, 4 ring, 353 Antecedent, 4 COPYRIGHTED MATERIALseparation, 228 Antichain, 183 , 229 Antisymmetric, 177 union, 228 Arbitrary, 88 Zermelo, 231 form, 21 Axiomatizable, finitely, 409 Assignment, 7 Associative, 34, 139, 199, 412 Basis case, 258 Assumption, 46 Bernstein, Felix, 301 Asymmetric, 177 Beth, 316 Atom, 3, 6 Biconditional, 5 Automorphism, 380 Biconditional proof, 107 435 AFirst Course in Mathematical and , First Edition. Michael L. O’Leary. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc. 436 INDEX

Bijection, 211 Complement, 128 Binary, 68 Complete, 56, 398 , 189 Complete theory, 423 operation, 198 Completeness , 61, 407 relation, 161 Gödel’s, 407 Binomial coefficient, 263 Completeness, real numbers, 275 Binomial theorem, 263 Complex number, 281 Boole, George, 117 Composite number, 107 Bound Composition, 163, 195 lower, 181 Compound, 4, 6 upper, 180 Concatenation, 178 Bound occurrence, 77 Conclusion, 22, 26 Burali-Forti paradox, 297 Conditional, 4 Burali-Forti theorem, 292 Conditional proof, 45 Congruent, 170 Cancellation, 248, 256, 414 Conjunct, 4 Candidate, 99 Conjunction, 4 Cantor–Schröder–Bernstein theorem, 301 Conjunction, inference rule, 25 Cantor, Georg, 117, 226, 229, 303, 313 Connective, 6 Cardinal, 307 Consequence, 22, 346, 352 large, 332 Consequent, 4 limit, 314 Consistency, relative, 407 regular, 328 Consistent, 52, 395 singular, 328 maximally, 53, 396 strongly inaccessible, 332 Constant , 68 successor, 314 Constructive dilemma, 25 weakly inaccessible, 331 Contingency, 16 Cardinality, 308 Continuous, 94 Cartesian n-space, 131 Continuum hypothesis, 313 Cartesian plane, 130 generalized, 314 Cartesian product, 130 , 16 Cases, 112 Contradiction, proof by, 47 Chain, 182 Contrapositive, 32, 103 elementary, 394 Contrapositive law, 34 of structures, 372 Converse, 32 Characteristic function, 304 Coordinates, 130 Choice axiom, 231, 235 Coordinatewise, 356 Choice function, 231 Copy, 214 , 420 Corollary, 20 Class (relation), 171 Corresponding occurrences, 77 Closed, 198 Countable, 310 Closed interval, 120 Counterexample, 102 Closed under deductions, 409 Cyclic group, 363, 365 , 190 Cofinal, 328 De Morgan, Augustus, 117 Cofinality, 328 De Morgan’s laws, 35, 137, 139 Cohen, Paul, 424 Decreasing, 184, 203 Coincidence, 349 Dedekind cut, 276 Combinatorics, 260 Dedekind, Richard, 229, 276 Common divisor, 140 Deduce, 26 Commutative, 34, 139, 199, 412 Deduction, 20 Commutative ring, 354 Deduction theorem, 42 Compactness theorem, 52, 415 Deductive logic, 2 Comparable, 181 Dense, 257 Compatible, 183 Denumerable, 310 INDEX 437

Descartes, René, 117 Even integer, 98 Destructive dilemma, 25 Excluded middle, 37 Diagonalization, 303 Exclusive or, 11 Diamond principle, 424 Existence, 104 Direct existential proof, 99 Existential formula, 72 Direct proof, 45 Existential generalization, 91 Discrete, 310 Existential instantiation, 91 Disjoint, 127, 155 Existential proof, 99, 106 pairwise, 155 Existential , 65 Disjunct, 4 Existential quantifier, 65 Disjunction, 4 Expansion, 352 Disjunctive normal form, 38 Exponentiation, 249, 321, 322 Disjunctive , 25 Exportation, 35 Distributive, 34, 139, 246, 323, 412 Extension, 197, 361 left, 320 elementary, 389 Divides, 98 Extensionality axiom, 227 Divisible, 98 Division algorithm, 185 Factor, 98, 110 polynomial, 110 Factorial, 242 Division ring, 357 False, 3 Divisor, 98 Family of sets, 148 common, 140 Fibonacci, 268 zero, 356 Fibonacci number, 269 Domain, 162, 334 Fibonacci sequence, 269 Dominate, 300 generalized, 273 Double , 35 Field, 357 Downward closed, 274 ordered, 69, 425 Downward Löwenheim–Skolem theorem, 415 Field theory, 69 Finite, 308 , 117 hereditarily, 417 Elementary chain, 394 Finitely axiomatizable, 409 Elementary equivalent, 387 First-order Elementary extension, 389 alphabet, 68 Elementary substructure, 389 formula, 73 Embedding, 214 language, 73 , 118, 141, 227 logic, 96 Empty set axiom, 227 , 26 Empty string, 6 Formation sequence, 7 Endpoint, 120 Formula, 71, 72 Equal, 118 first-order, 73 Equality, 194 second-order, 73 Equality , 226 Foundation axiom, 234 Equality symbol, 68 Fraenkel, Abraham, 229 Equinumerous, 298 Fraktur, 334, 428 Equivalence class, 171 Free occurrence, 77 Equivalence relation, 169 Free variable, 78 induced, 175 Frege, Gottlob, 24, 117 Equivalence rule, 110 Function, 189 Equivalence, logical, 31, 348 bijection, 211 Equivalent binary, 189 elementary, 387 characteristic, 304 pairwise, 109 choice, 231 Euclid, 19, 107, 265 continuous, 94 Euclid’s lemma, 265 decreasing, 203 Evaluation , 193 embedding, 214 438 INDEX

evaluation, 193 Half-open interval, 120 greatest integer, 192 Hartogs’ function, 327 homomorphism, 375 Hartogs’ theorem, 293 identity, 191 Hausdorff maximal principle, 237 inclusion, 203 Henkin, Leon, 399 increasing, 203 Henkin’s theorem, 406 injection, 206 Hereditarily finite sets, 417 inverse, 204 Hereditary set, 235 invertible, 204 Hierarchy, von Neumann, 417 isomorphism, 382 Hilbert, David, 226, 408 one-to-one, 206 Hilbert’s problems, 408 one-to-one correspondence, 211 Homomorphism, 375 onto, 208 group, 375 order-preserving, 212 ring, 375 periodic, 203 Hypothetical syllogism, 25 projection, 210 real-valued, 193 Ideal, 368 surjection, 208 improper, 368 unary, 189 left, 368 uniformly continuous, 94 maximal, 374 zero, 376 prime, 374 principal, 370 Function equality, 194 principal left, 370 Function notation, 190 proper, 368 Function symbol, 68 right, 368 Fundamental homomorphism theorem, 383 Idempotent laws, 139 Fundamental theorem of arithmetic, 271 Identity, 162, 199 additive, 199 Galois, Évariste, 342 multiplicative, 199 General linear group, 345 Identity map, 191 Generalization, 85 , 190, 208, 216 existential, 91 Implication, 4 universal, 88 Improper ideal, 368 Generalized continuum hypothesis, 314 Improper subgroup, 363 Generalized Fibonacci sequence, 273 Improper subring, 366 Generator, 363, 370 Improper subset, 136 Gödel, Kurt, 399, 407, 414, 423 Inaccessible cardinal, 331 Gödel’s completeness theorem, 407 Inclusion map, 203 Gödel’s incompleteness , 423 Inclusive or, 11 Golden ratio, 271 Incomparable, 181 Grammar, 6 Incompatible, 183 Greatest common divisor, 140 Incomplete theory, 423 Greatest element, 180 Incompleteness theorems, 423 Greatest integer function, 192 Inconsistent, 52, 395 Greatest lower bound, 181 Increasing, 184, 203 Grounded, 419 Independent, 408 Group, 342 Index, 148 abelian, 342 Index set, 148 cyclic, 363 Indexed, 149 general linear, 345 Indirect existential proof, 106 Klein-4, 344 Indirect proof, 47 simple, 363 Induced equivalence relation, 175 Group axioms, 340 Induced partition, 174 Group theory, 69 Induction Grouping symbol, 6 mathematical, 257 INDEX 439

on formulas, 336 König’s theorem, 330 on propositional forms, 59 Kuratowski, Kazimierz, 130, 232, 237 on terms, 335 strong, 268 Language, 73 transfinite, 283, 291 first order, 73 Induction hypothesis, 59, 258 Large cardinal, 332 Induction step, 258 , 37 Inductive logic, 1 Law of the excluded middle, 37 Inductive set, 238 Least element, 180 Infer, 24 Least upper bound, 180 Inference, 24 Left distributive, 320 Inference rule, 25 Left ideal, 368 Infinite, 308, 344 Left inverse, 215 Infinity axiom, 227 Left zero divisor, 356 Infinity symbol, 120 Leibniz, Gottfried, 117 Initial number, 327 Lemma, 20 Initial segment, 274 Leonardo of Pisa, 268 proper, 274 Lexicographical order, 187 Injection, 206 Liber abaci, 268 Instantiation, 85 Limit cardinal, 314 existential, 91 Limit ordinal, 290 universal, 87 Lindenbaum’s theorem, 397 Integers, 250 Linear combination, 186 Integral domain, 356 Linear order, 182 International Congress of Mathematicians, 408 Linearly ordered set, 182 , 335 Logic, 1 Intersection, 126, 152 deductive, 2 Interval, 120 first-order, 96 Interval notation, 120 inductive, 1 Introduction, 97 mathematical, 2 Invalid propositional, 20 semantically, 21 second-order, 96 syntactically, 21 Logic symbol, 67 Inverse, 166, 199 Logical implication, 21 additive, 199 Logical system, 20 function, 204 Logically equivalent, 31, 348 image, 216 Löwenheim, Leopold, 415 left, 215 Löwenheim’s theorem, 416 multiplicative, 199 Löwenheim–Skolem theorem right, 216 downward, 415 Inverse image, 216 upward, 416 Inverse relation, 166 Lower bound, 181 Inverse statement, 37 greatest, 181 Invertible function, 204 Łukasiewicz, Jan, 24, 71 Invertible matrix, 345 Irrational number, 128, 276 Mal’tsev, Anatolij, 414 Irreflexive, 177 Map, 190 Isomorphic, 212, 344, 380, 382 Martin’s axiom, 424 Isomorphism, 212, 380 Material equivalence, 13, 35 group, 382 Material implication, 12, 35 ring, 382 Mathematical induction, 257 strong, 268 Kernel, 379 , 2 Klein-4 group, 344, 363 Mathematics, 1 König, Julius, 301, 330 Mathesis universalis, 117 440 INDEX

Matrix, 345 partial, 178 identity, 345 well, 183 invertible, 345 Order isomorphism, 212 zero, 355 Order of connectives, 8, 9 Matrix addition, 355 Order of operations, 132 Matrix multiplication, 345 Order type, 212, 292 Maximal element, 181 Order-preserving, 212 Maximal ideal, 374 Ordered n-tuple, 131 Maximally consistent, 53, 396 Ordered field, 69, 425 Meaning, 334 , 130 Metatheory, 96 Ordinal, 286 Minimal element, 181 limit, 290 Mirimanoff, Dmitry, 229, 234 successor, 290 Model, 338, 339 Ordinal number, 286 Modern square of opposition, 86 Modulo, 170, 172 Pairing axiom, 228 Modus ponens, 25 Pairwise disjoint, 155 Modus tolens, 25 Pairwise equivalent, 109 Multiple, 98 Paragraph proof, 97 Multiplication, matrix, 345 Parsing tree, 6 Multiplicative identity, 199, 412 Partial order, 178 Multiplicative inverse, 176, 199 Partially ordered set, 178 Mutually exclusive, 127 Particular, 88 Partition, 173 N-ary, 68, 161, 189 induced, 174 Name, 334 Pascal’s identity, 263 Natural number, 238 Peano arithmetic, 69, 411 Necessary, 4, 109 Peano axioms, 410 Negated, 4 Peano, Giuseppe, 410 Negation, 4 Periodic, 203 Niece, 64, 161 Permutation, 260 Noncontradiction law, 37 Pigeonhole principle, 309 Nonstandard model, 411 Polynomial division algorithm, 110 Norway, 342 Poset, 178 Number, 256 Positive form, 86 standard, 411 Postulate, 20 Number theory, 69 Power, 415 Power set, 151 Occurrence, 65 Power set axiom, 228 Odd integer, 98 Pre-image, 190 One-to-one correspondence, 211 Predecessor, 237 One-to-one function, 206 Predicate, 64, 65 Onto function, 208 Prefix notation, 71 Open interval, 120 Premise, 22, 26 Operation, binary, 198 Prenex normal form, 96 associative, 199 Preserves, 212, 375 commutative, 199 Prime ideal, 374 Operation, set, 126 Prime number, 107 Or, 4 Prime power decomposition, 272 exclusive, 11 Principal ideal, 370 inclusive, 11 left, 370 Order, 344 Principal ideal domain, 372 increasing, 184 Projection, 210 lexicographical, 187 Proof methods linear, 182 biconditional, 107 INDEX 441

biconditional, short rule, 108 antisymmetric, 177 cases, 112 asymmetric, 177 conditional, 45 binary, 161 contradiction, 47 inverse, 166 counterexample, 102 irreflexive, 177 direct, 45 reflexive, 169 disjunctions, 111 symmetric, 169 equivalence rule, 110 transitive, 169 existence, 104 unary, 161 existential, direct, 99 Relation on, 161 existential, indirect, 106 Relation symbol, 68 indirect, 47 Relative consistency, 407 reductio ad absurdum, 47 Remainder, 110, 185 relative consistency, 407 Replacement axiom, 230 uniqueness, 104 Replacement rule, 34 universal, 98 Restriction, 197 Proof, formal, 26 Right ideal, 368 Proof, paragraph, 97 Right inverse, 216 Proof, two-column, 27 Right zero divisor, 356 Proper ideal, 368 Ring, 354 Proper initial segment, 274 commutative, 354 Proper subgroup, 364 division, 357 Proper subring, 366 field, 357 Proper subset, 136 integral domain, 356 Proposition, 3 principal ideal domain, 372 compound, 4 skew field, 357 universal, 66 with unity, 354 Proposition alphabet, 5 Ring axioms, 353 Propositional form, 6 Ring theory, 69 Propositional logic, 20 Roster, 118 Propositional variable, 6 Roster method, 118 Prove, 26 Russell’s paradox, 225 Pure set, 235 Russell, Bertrand, 225 Purely relational, 337 Satisfiable, 340, 352 Quantifier, 65, 66 Satisfy, 65, 338, 339 Quantifier negation, 86 Scheme, axiom, 228 Quantifier symbol, 68 Schröder, Ernst, 301 Quotient, 110, 185 Scope, 77 Quotient set, 172 Second-order alphabet, 73 Range, 162, 208 formula, 73 Rational numbers, 254 Second-order logic, 96 Ray, 120 Selector, 231 Real number, 276 Self-evident, 24 Real-valued function, 193 Semantically invalid, 21 , 242 Semantically valid, 21, 22 transfinite, 294 Semantics, 21 Recursive definition, 6, 242 Sentence, 83 Reduct, 352 Separation axioms, 228 Reductio ad absurdum, 47 Set, 117 Reflexive, 169 hereditary, 235 Regular cardinal, 328 Set difference, 127 Regularity axiom, 234 Set operation, 126 Relation, 161 Set theory, 68 442 INDEX

Short rule of biconditional proof, 108 Theory symbol, 68 Signature, 334 Tower, 233 Simple group, 363 Transcendental, 315 Simplification, 25 Transfinite induction, 283, 291 Simultaneous substitution, 81 Transfinite recursion, 294 Singleton, 118 Transitive, 169 Singular cardinal, 328 Transitive closure, 419 Skew field, 357 , 239 Skolem’s theorem, 416 Trichotomy, 183, 288 Skolem, Thoralf, 229, 234, 415 weak, 179 Sound, 56, 57, 398 Trichotomy law, 183 Soundness theorem, 57, 398 Trivial subgroup, 363 Square of opposition, modern, 86 Trivial subring, 366 Standard model, 411 True, 3 Standard number, 411 , 11 String, 5 , 3 Strong induction, 268 Two-column proof, 27 Strongly inaccessible, 332 Structure, 333 Unary, 68 pureley relational, 337 function, 189 Subgroup, 363 relation, 161 improper, 363 Undecidable, 313 proper, 364 Undefined, 190 trivial, 363 Uniformly continuous, 94 Subject, 64 Union, 126, 151 Subproof, 46 Union axiom, 228 Subring, 366 Unique, 104 improper, 366 Unique factorization theorem, 271 proper, 366 Unit, 357 trivial, 366 Unity, 354 Subset, 135 Universal formula, 72 Subset axioms, 229 Universal generalization, 88 Substitution, 64, 75, 78 Universal instantiation, 87 simultaneous, 81 Universal proof, 98 Substructure, 361 Universal proposition, 66 elementary, 389 Universal quantifier, 66 Successor, 237 Universe, 97, 141 Successor cardinal, 314 Upper bound, 180 Successor ordinal, 290 least, 180 Sufficient, 4, 109 Upward Löwenheim–Skolem theorem, 416 Surjection, 208 Urelement, 417 Symmetric, 169 Symmetric closure, 177 Valid, 347 Symmetric difference, 236 semantically, 21, 22 Syntactically invalid, 21 syntactically, 21, 27 Syntactically valid, 21, 27 Valuation, 10, 13 , 23 Valuation patterns, 15 Variable, 67 Tarski, Alfred, 336, 415 propositional, 6 Tarski–Vaught theorem, 390 Variable symbol, 67 Tautology, 16 Vaught, Robert, 336 Tautology rule, 35 , 127 Term, 70 Venn, John, 127 Theorem, 20, 27 Vertical line test, 189 Theory, 395 Von Neumann hierarchy, 417 INDEX 443

Von Neumann, John, 234, 238, 417

Weak-trichotomy law, 179 Weakly inaccessible, 331 Well-defined, 191 Well-ordered set, 183 Well-ordering theorem, 293 Whitehead, John H. C., 424 Whitehead problem, 424 Witness, 91 Witness set, 399 Write, 23

Zermelo axioms, 235 Zermelo’s axiom, 231 Zermelo’s theorem, 295 Zermelo, Ernst, 225, 234, 293, 417 Zermelo–Fraenkel axioms, 235 Zero, 102, 110 Zero divisor, 356 left, 356 right, 356 Zero map, 376 Zero matrix, 355 Zorn’s lemma, 232 Zorn, Max, 232