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Copyrighted Material INDEX Abel, Niels, 342 Axiom scheme, 228 Abelian group, 342 Axiom(s), 24 Absolute value, 116 choice, 231, 235 Abstraction, 122 empty set, 227 Addition, inference rule, 25 equality, 226 Addition, matrix, 355 extensionality, 227 Additive identity, 199, 412 foundation, 234 Additive inverse, 199 Frege–Łukasiewicz, 24 Aleph, 313 group, 340 Algebraic, 315 paring, 228 Alphabet, 5 power set, 228 first-order, 68 regularity, 234 second-order, 73 replacement, 230 And, 4 ring, 353 Antecedent, 4 COPYRIGHTED MATERIALseparation, 228 Antichain, 183 subset, 229 Antisymmetric, 177 union, 228 Arbitrary, 88 Zermelo, 231 Argument 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 Logic and Set Theory, 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 function, 189 Complete theory, 423 operation, 198 Completeness theorem, 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 symbol, 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 Contradiction, 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 Class, 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 Codomain, 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 proposition, 65 Disjunct, 4 Existential quantifier, 65 Disjunction, 4 Expansion, 352 Disjunctive normal form, 38 Exponentiation, 249, 321, 322 Disjunctive syllogism, 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 negation, 35 Field, 357 Downward closed, 274 ordered, 69, 425 Downward Löwenheim–Skolem theorem, 415 Field theory, 69 Finite, 308 Element, 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 Empty set, 118, 141, 227 logic, 96 Empty set axiom, 227 Formal proof, 26 Empty string, 6 Formation sequence, 7 Endpoint, 120 Formula, 71, 72 Equal, 118 first-order, 73 Equality, 194 second-order, 73 Equality axioms, 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 map, 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 Image, 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 theorems, 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 Law of noncontradiction, 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 Interpretation, 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 Mathematical logic, 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
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