INDEX
Abel, Niels, 342
Abelian group, 342
Absolute value, 116
Abstraction, 122
Addition, inference rule, 25
Addition, matrix, 355
Additive inverse, 199
Aleph, 313
Algebraic, 315
Alphabet, 5
first-order, 68
second-order, 73
And, 4
Antecedent, 4
Antichain, 183
Antisymmetric, 177
Arbitrary, 88
Argument form, 21
Assignment, 7
Associative, 34, 139, 199, 412
Assumption, 46
Asymmetric, 177
Automorphism, 380
Axiom scheme, 228
Axiom(s), 24
empty set, 227
equality, 226
extensionality, 227
foundation, 234
Frege–Łukasiewicz, 24
group, 340
paring, 228
power set, 228
regularity, 234
replacement, 230
ring, 353
separation, 228
subset, 229
union, 228
Zermelo, 231
Axiomatizable, finitely, 409
Basis case, 258
Bernstein, Felix, 301
Beth, 316
Biconditional, 5
Biconditional proof, 107
Bijection, 211
Binary, 68
function, 189
operation, 198
relation, 161
Binomial coefficient, 263
Binomial theorem, 263
Boole, George, 117
Bound
lower, 181
upper, 180
Bound occurrence, 77
Burali-Forti paradox, 297
Burali-Forti theorem, 292
Candidate, 99
Cantor–Schröder–Bernstein theorem, 301
Cantor, Georg, 117, 226, 229, 303, 313
Cardinal, 307
large, 332
limit, 314
regular, 328
singular, 328
strongly inaccessible, 332
successor, 314
weakly inaccessible, 331
Cardinality, 308
Cartesian n-space, 131
Cartesian plane, 130
Cartesian product, 130
Cases, 112
Chain, 182
elementary, 394
of structures, 372
Characteristic function, 304
Choice function, 231
Class, 420
Class (relation), 171
Closed, 198
Closed interval, 120
Closed under deductions, 409
Codomain, 190
Cofinal, 328
Cofinality, 328
Cohen, Paul, 424
Coincidence, 349
Combinatorics, 260
Common divisor, 140
Commutative, 34, 139, 199, 412
Commutative ring, 354
Comparable, 181
Compatible, 183
Complement, 128
Complete theory, 423
Gödel's, 407
Completeness, real numbers, 275
Complex number, 281
Composite number, 107
Concatenation, 178
Conditional, 4
Conditional proof, 45
Congruent, 170
Conjunct, 4
Conjunction, 4
Conjunction, inference rule, 25
Connective, 6
Consequent, 4
Consistency, relative, 407
Constant symbol, 68
Constructive dilemma, 25
Contingency, 16
Continuous, 94
Continuum hypothesis, 313
generalized, 314
Contradiction, 16
Contradiction, proof by, 47
Contrapositive law, 34
Converse, 32
Coordinates, 130
Coordinatewise, 356
Copy, 214
Corollary, 20
Corresponding occurrences, 77
Countable, 310
Counterexample, 102
De Morgan, Augustus, 117
De Morgan's laws, 35, 137, 139
Dedekind cut, 276
Deduce, 26
Deduction, 20
Deduction theorem, 42
Deductive logic, 2
Dense, 257
Denumerable, 310
Descartes, René, 117
Destructive dilemma, 25
Diagonalization, 303
Diamond principle, 424
Direct existential proof, 99
Direct proof, 45
Discrete, 310
pairwise, 155
Disjunct, 4
Disjunction, 4
Disjunctive normal form, 38
Disjunctive syllogism, 25
Distributive, 34, 139, 246, 323, 412
left, 320
Divides, 98
Divisible, 98
Division algorithm, 185
polynomial, 110
Division ring, 357
Divisor, 98
common, 140
zero, 356
Dominate, 300
Double negation, 35
Downward closed, 274
Downward Löwenheim–Skolem theorem, 415
Element, 117
Elementary chain, 394
Elementary equivalent, 387
Elementary extension, 389
Elementary substructure, 389
Embedding, 214
Empty set axiom, 227
Empty string, 6
Endpoint, 120
Equal, 118
Equality, 194
Equality axioms, 226
Equality symbol, 68
Equinumerous, 298
Equivalence class, 171
Equivalence relation, 169
induced, 175
Equivalence rule, 110
Equivalent
elementary, 387
pairwise, 109
Euclid's lemma, 265
Evaluation map, 193
Even integer, 98
Excluded middle, 37
Exclusive or, 11
Existence, 104
Existential formula, 72
Existential generalization, 91
Existential instantiation, 91
Existential proposition, 65
Existential quantifier, 65
Expansion, 352
Exportation, 35
elementary, 389
Extensionality axiom, 227
Factorial, 242
False, 3
Family of sets, 148
Fibonacci, 268
Fibonacci number, 269
Fibonacci sequence, 269
generalized, 273
Field, 357
Field theory, 69
Finite, 308
hereditarily, 417
Finitely axiomatizable, 409
First-order
alphabet, 68
formula, 73
language, 73
logic, 96
Formal proof, 26
Formation sequence, 7
first-order, 73
second-order, 73
Foundation axiom, 234
Fraenkel, Abraham, 229
Free occurrence, 77
Free variable, 78
Function, 189
bijection, 211
binary, 189
characteristic, 304
choice, 231
continuous, 94
decreasing, 203
embedding, 214
evaluation, 193
greatest integer, 192
homomorphism, 375
identity, 191
inclusion, 203
increasing, 203
injection, 206
inverse, 204
invertible, 204
isomorphism, 382
one-to-one, 206
one-to-one correspondence, 211
onto, 208
order-preserving, 212
periodic, 203
projection, 210
real-valued, 193
surjection, 208
unary, 189
uniformly continuous, 94
zero, 376
Function equality, 194
Function notation, 190
Function symbol, 68
Fundamental homomorphism theorem, 383
Fundamental theorem of arithmetic, 271
Galois, Évariste, 342
General linear group, 345
Generalization, 85
existential, 91
universal, 88
Generalized continuum hypothesis, 314
Generalized Fibonacci sequence, 273
Gödel, Kurt, 399, 407, 414, 423
Gödel's completeness theorem, 407
Gödel's incompleteness theorems, 423
Golden ratio, 271
Grammar, 6
Greatest common divisor, 140
Greatest element, 180
Greatest integer function, 192
Greatest lower bound, 181
Grounded, 419
Group, 342
abelian, 342
cyclic, 363
general linear, 345
Klein-4, 344
simple, 363
Group axioms, 340
Group theory, 69
Grouping symbol, 6
Half-open interval, 120
Hartogs' function, 327
Hartogs' theorem, 293
Hausdorff maximal principle, 237
Henkin, Leon, 399
Henkin's theorem, 406
Hereditarily finite sets, 417
Hereditary set, 235
Hierarchy, von Neumann, 417
Hilbert's problems, 408
Homomorphism, 375
group, 375
ring, 375
Hypothetical syllogism, 25
Ideal, 368
improper, 368
left, 368
maximal, 374
prime, 374
principal, 370
principal left, 370
proper, 368
right, 368
Idempotent laws, 139
additive, 199
multiplicative, 199
Identity map, 191
Implication, 4
Improper ideal, 368
Improper subgroup, 363
Improper subring, 366
Improper subset, 136
Inaccessible cardinal, 331
Inclusion map, 203
Inclusive or, 11
Incomparable, 181
Incompatible, 183
Incomplete theory, 423
Incompleteness theorems, 423
Independent, 408
Index, 148
Index set, 148
Indexed, 149
Indirect existential proof, 106
Indirect proof, 47
Induced equivalence relation, 175
Induced partition, 174
Induction
mathematical, 257
on formulas, 336
on propositional forms, 59
on terms, 335
strong, 268
Induction step, 258
Inductive logic, 1
Inductive set, 238
Infer, 24
Inference, 24
Inference rule, 25
Infinity axiom, 227
Infinity symbol, 120
Initial number, 327
Initial segment, 274
proper, 274
Injection, 206
Instantiation, 85
existential, 91
universal, 87
Integers, 250
Integral domain, 356
International Congress of Mathematicians, 408
Interpretation, 335
Interval, 120
Interval notation, 120
Introduction, 97
Invalid
semantically, 21
syntactically, 21
additive, 199
function, 204
image, 216
left, 215
multiplicative, 199
right, 216
Inverse image, 216
Inverse relation, 166
Inverse statement, 37
Invertible function, 204
Invertible matrix, 345
Irreflexive, 177
Isomorphic, 212, 344, 380, 382
group, 382
ring, 382
Kernel, 379
König's theorem, 330
Kuratowski, Kazimierz, 130, 232, 237
Language, 73
first order, 73
Large cardinal, 332
Law of noncontradiction, 37
Law of the excluded middle, 37
Least element, 180
Least upper bound, 180
Left distributive, 320
Left ideal, 368
Left inverse, 215
Left zero divisor, 356
Leibniz, Gottfried, 117
Lemma, 20
Leonardo of Pisa, 268
Lexicographical order, 187
Liber abaci, 268
Limit cardinal, 314
Limit ordinal, 290
Lindenbaum's theorem, 397
Linear combination, 186
Linear order, 182
Linearly ordered set, 182
Logic, 1
deductive, 2
first-order, 96
inductive, 1
mathematical, 2
propositional, 20
second-order, 96
Logic symbol, 67
Logical implication, 21
Logical system, 20
Löwenheim, Leopold, 415
Löwenheim's theorem, 416
Löwenheim–Skolem theorem
downward, 415
upward, 416
Lower bound, 181
greatest, 181
Mal'tsev, Anatolij, 414
Map, 190
Martin's axiom, 424
Mathematical induction, 257
strong, 268
Mathematical logic, 2
Mathematics, 1
Mathesis universalis, 117
Matrix, 345
identity, 345
invertible, 345
zero, 355
Matrix addition, 355
Matrix multiplication, 345
Maximal element, 181
Maximal ideal, 374
Meaning, 334
Metatheory, 96
Minimal element, 181
Modern square of opposition, 86
Modus ponens, 25
Modus tolens, 25
Multiple, 98
Multiplication, matrix, 345
Multiplicative identity, 199, 412
Multiplicative inverse, 176, 199
Mutually exclusive, 127
Name, 334
Natural number, 238
Negated, 4
Negation, 4
Noncontradiction law, 37
Nonstandard model, 411
Norway, 342
Number, 256
standard, 411
Number theory, 69
Occurrence, 65
Odd integer, 98
One-to-one correspondence, 211
One-to-one function, 206
Onto function, 208
Open interval, 120
Operation, binary, 198
associative, 199
commutative, 199
Operation, set, 126
Or, 4
exclusive, 11
inclusive, 11
Order, 344
increasing, 184
lexicographical, 187
linear, 182
partial, 178
well, 183
Order isomorphism, 212
Order of operations, 132
Order-preserving, 212
Ordered n-tuple, 131
Ordered pair, 130
Ordinal, 286
limit, 290
successor, 290
Ordinal number, 286
Pairing axiom, 228
Pairwise disjoint, 155
Pairwise equivalent, 109
Paragraph proof, 97
Parsing tree, 6
Partial order, 178
Partially ordered set, 178
Particular, 88
Partition, 173
induced, 174
Pascal's identity, 263
Peano axioms, 410
Peano, Giuseppe, 410
Periodic, 203
Permutation, 260
Pigeonhole principle, 309
Polynomial division algorithm, 110
Poset, 178
Positive form, 86
Postulate, 20
Power, 415
Power set, 151
Power set axiom, 228
Pre-image, 190
Predecessor, 237
Prefix notation, 71
Prenex normal form, 96
Prime ideal, 374
Prime number, 107
Prime power decomposition, 272
Principal ideal, 370
left, 370
Principal ideal domain, 372
Projection, 210
Proof methods
biconditional, 107
biconditional, short rule, 108
cases, 112
conditional, 45
contradiction, 47
counterexample, 102
direct, 45
disjunctions, 111
equivalence rule, 110
existence, 104
existential, direct, 99
existential, indirect, 106
indirect, 47
reductio ad absurdum, 47
relative consistency, 407
uniqueness, 104
universal, 98
Proof, formal, 26
Proof, paragraph, 97
Proof, two-column, 27
Proper ideal, 368
Proper initial segment, 274
Proper subgroup, 364
Proper subring, 366
Proper subset, 136
Proposition, 3
compound, 4
universal, 66
Proposition alphabet, 5
Propositional form, 6
Propositional logic, 20
Propositional variable, 6
Prove, 26
Pure set, 235
Purely relational, 337
Quantifier negation, 86
Quantifier symbol, 68
Quotient set, 172
Rational numbers, 254
Ray, 120
Real number, 276
Real-valued function, 193
Recursion, 242
transfinite, 294
Reduct, 352
Reductio ad absurdum, 47
Reflexive, 169
Regular cardinal, 328
Regularity axiom, 234
Relation, 161
antisymmetric, 177
asymmetric, 177
binary, 161
inverse, 166
irreflexive, 177
reflexive, 169
symmetric, 169
transitive, 169
unary, 161
Relation on, 161
Relation symbol, 68
Relative consistency, 407
Replacement axiom, 230
Replacement rule, 34
Restriction, 197
Right ideal, 368
Right inverse, 216
Right zero divisor, 356
Ring, 354
commutative, 354
division, 357
field, 357
integral domain, 356
principal ideal domain, 372
skew field, 357
with unity, 354
Ring axioms, 353
Ring theory, 69
Roster, 118
Roster method, 118
Russell's paradox, 225
Russell, Bertrand, 225
Scheme, axiom, 228
Schröder, Ernst, 301
Scope, 77
Second-order
alphabet, 73
formula, 73
Second-order logic, 96
Selector, 231
Self-evident, 24
Semantically invalid, 21
Semantics, 21
Sentence, 83
Separation axioms, 228
Set, 117
hereditary, 235
Set difference, 127
Set operation, 126
Set theory, 68
Short rule of biconditional proof, 108
Signature, 334
Simple group, 363
Simplification, 25
Simultaneous substitution, 81
Singleton, 118
Singular cardinal, 328
Skew field, 357
Skolem's theorem, 416
Skolem, Thoralf, 229, 234, 415
Square of opposition, modern, 86
Standard model, 411
Standard number, 411
String, 5
Strong induction, 268
Strongly inaccessible, 332
Structure, 333
pureley relational, 337
Subgroup, 363
improper, 363
proper, 364
trivial, 363
Subject, 64
Subproof, 46
Subring, 366
improper, 366
proper, 366
trivial, 366
Subset, 135
Subset axioms, 229
simultaneous, 81
Substructure, 361
elementary, 389
Successor, 237
Successor cardinal, 314
Successor ordinal, 290
Surjection, 208
Symmetric, 169
Symmetric closure, 177
Symmetric difference, 236
Syntactically invalid, 21
Syntax, 23
Tarski–Vaught theorem, 390
Tautology, 16
Tautology rule, 35
Term, 70
Theory, 395
Theory symbol, 68
Tower, 233
Transcendental, 315
Transfinite induction, 283, 291
Transfinite recursion, 294
Transitive, 169
Transitive closure, 419
Transitive set, 239
weak, 179
Trichotomy law, 183
Trivial subgroup, 363
Trivial subring, 366
True, 3
Truth table, 11
Truth value, 3
Two-column proof, 27
Unary, 68
function, 189
relation, 161
Undecidable, 313
Undefined, 190
Uniformly continuous, 94
Union axiom, 228
Unique, 104
Unique factorization theorem, 271
Unit, 357
Unity, 354
Universal formula, 72
Universal generalization, 88
Universal instantiation, 87
Universal proof, 98
Universal proposition, 66
Universal quantifier, 66
Upper bound, 180
least, 180
Upward Löwenheim–Skolem theorem, 416
Urelement, 417
Valid, 347
Valuation patterns, 15
Variable, 67
propositional, 6
Variable symbol, 67
Vaught, Robert, 336
Venn diagram, 127
Venn, John, 127
Vertical line test, 189
Von Neumann hierarchy, 417
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 divisor, 356
left, 356
right, 356
Zero map, 376
Zero matrix, 355
Zorn's lemma, 232
Zorn, Max, 232