Index of terminology

The reference numbers indicate the chapter, section, and sub-section (or exercise) in that order. (R refers to Summary of Results.)

Adjunction of a greatest element to an ordered set : III.1.7 Adjunction (of one set to another) : R.5.4 Agreeing on a set (functions) : II.3.5 Agreement (of two functions on a set) : R.2.14 Aleph : III.6. Ex. 10 Antecedent assemblies : I. App. 4 Antidirected (ordered set) : III.1. Ex. 23 Applying the results of one theory in another : I.2.4 Argument : R.1.2 Assembly : I.1.1     antecedent : I. App. 4     balanced : I. App. 4     of the first (second) species : I.1.3     perfectly balanced : I. App. 4 Associated equivalence relation : II.6.2 Associativity criterion for product structures : IV.2.4     of the intersection of a family of sets : R.4.8     of the product of a family of sets : R.4.11     of the union and intersection of two sets : R.1.14     of the union of a family of sets : R.4.3 Automorphism : IV.1.5, R.8.6 Auxiliary base sets : IV.1.3,1.4     constant, method of : I.3.3     hypothesis, method of : I.3.3 Axiom, explicit : I.2.1     implicit : I.2.1     of choice : R.4.10     of extent : II.1.3     of infinity : III.6.1

Axiom of the ordered pair : II.2.1 of the set of subsets : II.5.1 of the set of two elements : II.1.5 Axioms, equivalent : R.8.4 of structures of the same species : R.8.2 Axiom(s) of a species of structures : IV.1.4

Balanced assembly : I. App. 4     word : I. App. 2 Base of an expansion, or of a system of numeration : III.5.7     of a scale of sets : R.8.1     sets (of an echelon construction) : IV.1.1     sets, auxiliary : IV.1.3, 1.4     sets, principal : IV.1.3, 1.4 Bijection : II.3.7, R.2.9 Bijective mapping : II.3.7, R.2.9 Binomial coefficient : III.5.8 Boolean lattice : III.1. Ex. 17 Bound, greatest lower and least upper (of a set or a mapping) : III.1.9, R.6.7     least upper (of a set of cardinals) : III.3.2     lower : III.1.8, R.6.7     strict upper : III.2.4     upper : III.1.8, R.6.7 Bounded, bounded above, bounded below (set or mapping) : III.1.8, R.6.7 Branched (ordered set) : III.1. Ex. 24

Canonical decomposition of a function : II.6.5, R.5.3 Canonical extension, of a correspondence to sets of subsets : II.5.1     of a family of functions to the product sets : II.5.7     of mappings : IV.1.2     of two functions to the product sets : II.3.9     signed : IV.2. Ex. 1 Canonical injection : II.3.7 Canonical mapping, IV, App. 4     of a subset of E into E : II.3.7, R.2.3     of $A^{B\times C}$ onto $(A^B)^C$ : II.5.2, R.4.14     of $G$ onto $\overset{-1}{G}$ ($G$ a graph) : II.3.7     of $F^E$ onto $\mathfrak{F}(E,F)$ : II.5.2     of $\mathfrak{F}(B\times C,A)$ onto $\mathfrak{F}(B,\mathfrak{F}(C,A))$ : II.5.2     of $\displaystyle\prod_{\iota\in\{\alpha\}} X_\iota$ onto $X_\alpha$ : II.5.3     of $\displaystyle\prod_{\iota\in\{\alpha,\beta\}} X_\iota$ onto $X_\alpha\times X_\beta$ : II.5.3

Canonical mapping of $\prod_{\iota\in I} X_\iota$ onto $\prod_{\lambda\in L}\left(\prod_{\iota\in J_\lambda} X_\iota\right)$ : II.5.5, R.4.11

of $\prod_{\iota\in I} X_\iota$ onto $\left(\prod_{\iota\in J_\alpha} X_\iota\right)\times\left(\prod_{\iota\in J_\beta} X_\iota\right)$ (($J_\alpha,J_\beta$) a partition of $I$) : II.5.5

of $\prod_{\iota\in\{\alpha,\beta,\gamma\}} X_\iota$ onto $X_\alpha\times X_\beta\times X_\gamma$ : II.5.5

of $\prod_{\iota\in I} X_\iota^E$ onto $\left(\prod_{\iota\in I} X_\iota\right)^E$ : II.5.7, R.4.13

of $E$ onto $E/R$ : II.6.2, R.5.2

of $A/R_A$ onto $f\langle A\rangle$ : II.6.6, R.5.5

of $(E/S)/(R/S)$ onto $E/R$ : II.6.7, R.5.9

of $(E\times E')/(R\times R')$ onto $(E/R)\times(E'/R')$ : II.6.8, R.5.10

into a direct limit of a direct limit obtained by restriction of the index set : III.7.6

of $E_\beta$ into $\overrightarrow{\lim} E_\alpha$ : III.7.5, R.6.13

of an inverse limit into an inverse limit obtained by restriction of the index set : III.7.1

of $\lim E_\alpha$ into $E_\beta$ : III.7.1, R.6.14

of $\widetilde{E}\times F$ onto $F\times E$ : R.3.4

of $E\times F\times G$ onto $(E\times F)\times G$, etc. : R.3.12

of $F^A$ onto $\prod_{\iota\in I}F^{A_\iota}$ (where $A=\bigcup_{\iota\in I}A_\iota$, and $A_\iota\cap A_\gamma=\varnothing$ whenever $\iota\ne\gamma$) : R.4.15

symmetry : R.3.4

Cantor’s theorem : III.3.6

Cardinal, dominant : III.6.Ex.21 finite : III.4.1 inaccessible : III.6.Ex.22 product : III.3.3 regular : III.6.Ex.17 singular : III.6.Ex.17 strongly inaccessible : III.6.Ex.22

Cardinal of a set : III.3.1

Cardinal sum : III.3.3

Chain of an element in an ordered set (with respect to a mapping) : III.2.Ex.6

Characteristic function of a subset of a set : III.5.5

Characterization, typical : IV.1.4

Choice, axiom of : R.4.9

Class, equivalence : II.6.2, R.5.2 of objects equivalent to $x$ (with respect to an equivalence relation) : II.6.9

Closed interval : III.1.13, R.6.4

Closure (mapping of an ordered set into itself) : III.1.Ex.13

Coarser equivalence relation : II.6.7 preordering : III.1.4 structure : IV.2.2 Coefficient, binomial : III.5.8 Cofinal subset : III.1.7, R.6.5 Coincidence (of two functions on a set) : II.3.5, R.2.14 Coinitial subset : III.1.7, R.6.5 Collectivizing relation : II.1.4 Commutativity (of union and intersection) : R.1.14 Comparable elements : III.1.12 structures : IV.2.2 Compatibility, of a function with two equivalence relations : II.6.5, R.5.8 of a relation with an equivalence relation : II.6.5, R.5.7 Complement of a set : II.1.7, R.1.7 Complete lattice : III.1.Ex.11 Complete solution : I.5.2 Completely ramified (ordered set) : III.2.Ex.8 Completion of an ordered set : III.1.Ex.15 Composition, of mappings : R.2.11 of sets : R.3.10 of two correspondences : II.3.3 of two graphs : II.3.3 Conjunction of two relations : I.3.4 Connected components of a set with respect to a relation : II.6.Ex.10 Constant, auxiliary : I.3.3 function or mapping : II.3.4, R.2.3 of a theory : I.2.1 Constituents of a set with respect to a relation : II.6.Ex.11 Construction, echelon : IV.1.1 formative : I.1.3 Contained in a set : II.1.2, R.1.12 Continuum hypothesis : III.6.4 Continuum, power of : III.6.4 Contradictory axioms : R.8.6 theory : I.2.2 Contravariant signed echelon type : IV.2.Ex.1 Coordinate (first, second) of an ordered pair : II.2.1 Coordinate function (first, second) : II.3.6 of index $i$ : II.5.3 Coordinate functions, on a product of a family of sets : R.4.11 on a product of several sets : R.3.12 on a product of two sets : R.3.1 Corollary : I.2.2 Correspondence, between two sets : II.3.1

Correspondence defined at an object $x$ : II.3.1     defined by a relation : II.3.1     inverse : II.3.2     one-to-one : II.3.7, R.2.9 Correspondences, composition of : II.3.3 Countable set : III.6.4, R.7.7 Countable union, intersection, product : R.7.9 Covariant signed echelon type : IV.2.Ex.1 Covering : II.4.6, R.4.4     finer : II.4.6     of a set : II.4.6 Criterion, deductive : I.2.2     formative : I.1.4     of deduction : I.3.3     of substitution : I.1.2 Critical ordinal : III.6.Ex.13

Decent set : III.2.Ex.20 Decimal system : III.5.7 Decomposition, canonical : II.6.5, R.5.3 Decreasing family of subsets : III.1.5, R.6.12 Decreasing mapping : III.1.5, R.6.12 Deduced (structure) : IV.1.6 Deduction, criterion of : I.3.3     procedure : IV.1.6 Deductive criterion : I.2.2 Definition : I.1.1 Degree of disjointness of a covering : III.6.Ex.25 Demonstrative text : I.2.2 Descending induction : III.4.3 Diagonal, of $A \times A$ : II.3.3, R.3.4     of $E^I$ : II.5.3 Diagonal mapping, of $A$ into $A \times A$ : II.3.7, R.3.4     of $E$ into $E^I$ : II.5.3 Diagrams : R.2.2 Difference of two integers : III.5.2 Different from : I.5.1 Digit : III.5.7 Direct image of a structure : IV.2.6 Direct limit, of a direct system of mappings : III.7.6, R.6.13     of a direct system of sets : III.7.5, R.6.13 Direct system, of mappings : III.7.6, R.6.13     of sets : III.7.5, R.6.13     of subsets : III.7.6

Directed (left, right) : III.1.10, R.6.8  with respect to the relation $\leq$ : III.1.10 Disjoint segments : I.App.1  sets : II.4.7, R.1.13 Disjunction, of cases, method of : I.3.3  of two relations : I.1.3 Distributive lattice : III.1.Ex.16 Distributivity, of union and intersection of a family of sets : R.4.3, 4.8  of union and intersection of two sets : R.1.14 Divergent mapping (of an ordinal into itself) : III.6.Ex.23 Divisible by an integer : III.5.6 Division, Euclidean : III.5.6  of an integer : III.5.6 Domain, of a correspondence : II.3.1  of a graph : II.3.1 Dominant cardinal : III.6.Ex.21 Double family : II.3.4 Double sequence : III.6.1, R.7.8 Dual formulae : R.1.15, R.4.7 Duality rule : R.1.15, R.4.7 Dyadic system : III.5.7

Echelon, echelon construction, echelon construction scheme : IV.1.1  signed : IV.2.Ex.1  type : IV.1.Ex.1 Element : II.1.1  fixed (under a mapping or set of mappings) : R.2.3  generic : R.1.2  greatest : III.1.3, R.6.5  invariant (under a mapping or set of mappings) : R.2.3  least : III.1.3, R.6.5  maximal : III.1.6, R.6.6  minimal : III.1.6, R.6.6 Elements, comparable : III.1.12 Empty function : II.3.4 Empty set : II.1.7, R.1.8 Empty word : I.App.1 Endowed with a structure : IV.1.4 Endpoints of an interval : III.1.13, R.6.4 Equal : I.5.1 Equalitarian theory : I.5.1 Equality, relation of : I.5.1, R.1.6 Equation : I.5.2 Equipotent sets : III.3.1, R.7.1

Equivalence, on a set : II.6.1 Equivalence class : II.6.2, R.5.2 Equivalence relation : II.6.1, R.5.2   associated with a function : II.6.2   coarser : II.6.7   finer : II.6.7   induced : II.6.6   on a set : II.6.1   product : II.6.8   quotient : II.6.7 Equivalent axioms : R.8.4 Equivalent elements : II.6.1 Equivalent powers : R.7.2 Equivalent relations : I.3.5, R.1.3 Equivalent species of structures : IV.1.7 Equivalent theories : I.2.4 Euclidean division : III.5.6 Even integer : III.5.6 Existential quantifier : I.4.1 Expansion of an integer to base $b$ : III.5.7 Explicit axiom : I.2.1 Exponentiation : R.4.9 Extension, canonical : IV.1.2   inverse (of a mapping to sets of subsets) : R.2.6   of a correspondence or mapping to sets of subsets : II.5.1, R.2.4   of a family of functions to product sets : II.5.7   of a mapping : II.3.5, R.2.13   of an ordering, order relation, preordering, preorder relation: III.1.4, R.6.1   of several mappings to the product sets : R.3.14   of two functions to product sets : II.3.9 Extent, axiom of : II.1.3

Factor, of an integer : III.5.6   of a product : II.2.2 and II.5.3 Factors, of a product set : R.3.1, 4.9 Factorial $n$ : III.5.8 Factorization, canonical : R.5.3   of a mapping : R.2.11 False relation : I.2.2 Family : II.3.4   double : II.3.4   finite : III.4.1   of elements of a set : II.3.4, R.2.14   of mutually disjoint sets : II.4.7

Family of subsets, decreasing (increasing) : III.1.5     of subsets of a set : II.3.4 Final character (of a totally ordered set) : III.6.Ex.16 Final segment of a word : I.App.1 Final structure : IV.2.5 Finer covering : II.4.6 Finer preordering : III.1.4 Finer relation : II.6.7 Finer structure : IV.2.2 Finite cardinal : III.4.1 Finite character (property of, set of subsets of) : III.4.5, R.6.11 Finite family : III.4.1 Finite sequence, finite sequence of elements of a set : III.5.4, R.7.8 Finite set : IV.4.1 First coordinate of an ordered pair : II.2.1 First projection of a graph : II.3.1 First species : I.1.3 First term of a sequence : III.5.4 Fixed element : II.3.4, R.2.3 Formative construction : I.1.3     criterion : I.1.4 Free subset of an ordered set : III.1.1.Ex.5 Function : see mapping     characteristic : III.5.5     coordinate : II.3.6, II.5.3     defined on A with values in B : II.3.4     not depending on x : II.3.9     of several variables : R.3.13     of two arguments : II.3.9 Functional graph : II.3.4 Functional relation : I.5.3, R.2.1 Functional symbol : I.5.3

General solution : I.5.2 General term (of a sequence) : R.7.8 Generalized continuum hypothesis : III.6.4 Generic element (of a set) : R.1.2 Generic structure : IV.1.4 Graph : II.3.1     functional : II.3.4     inverse : II.3.2     of a correspondence : II.3.1     of a mapping : R.3.5     of a relation : II.3.1, R.3.2

Graph symmetric : II.3.2 Graphs, composition of : II.3.3 Greater : III.1.3, R.6.3 Greatest element of an ordered set : III.1.7, R.6.5 Greatest lower bound : III.1.9, R.6.7

Half-open interval : III.1.1.13, R.6.4 Hypothesis, auxiliary : I.3.3  continuum : III.6.4  generalized continuum : III.6.4  inductive : III.2.2, III.4.3

Identification : R.8.5 Identity : R.1.3 Identity correspondence : II.3.3 Identity mapping : II.3.4, R.2.3 Image, direct (of a structure) : IV.2.6  inverse (of a set under a mapping) : R.2.6  inverse (of a structure) : IV.2.4  of a covering : II.4.6  of an equivalence relation : II.6.6  of a function (by abuse of language) : R.2.4  of a set : II.3.2  of a set under a correspondence : II.3.1  of a set under a function : II.3.4, R.2.4  of a set under a graph : II.3.1 Implicit axiom : I.2.1 Imply : R.1.3 Inaccessible cardinal : III.6.Ex.22 Inaccessible initial ordinal : III.6.Ex.16 Inclusion relation : II.1.2, III.1.1, R.1.2 Inconsistent axioms : R.8.6 Increasing family of subsets : III.1.5, R.6.12 Increasing mapping : III.1.5, R.6.12 Indecomposable ordinal : III.2.Ex.16 Independent (of other axioms) : I.2.Ex.1 Index set of a family : II.3.4, R.2.2 Indicial notation : R.2.2 Induced equivalence relation : R.5.5 Induced mapping : II.6.5 Induced ordering, order relation, preordering, preorder relation : III.1.4, R.6.1 Induced relation : II.6.3 Induced structure : IV.2.4

Induction, descending : III.4.3 Noetherian : III.6.5 principle of : III.2.2, III.4.3 restricted : III.4.3 starting at k : III.4.3 transfinite : III.2.2 Inductive hypothesis : III.2.2, III.4.3 Inductive set : III.2.4, R.6.9 Infimum : III.1.9, R.6.7 Infinite sequence : III.6.1, R.7.8 Infinite set : III.6.1 Infinity, axiom of : III.6.1 Initial ordinal : III.6.Ex.10 Initial segment of a word : I.App.1 Initial structure : IV.2.3 Injection : II.3.7, R.2.8 canonical : II.3.7 Injective mapping : II.3.7, R.2.8 Integer, natural : III.4.1 Integral part of the quotient of two integers : III.5.6 Intersecting subsets : R.1.13 Intersection, countable : R.7.9 of a family of sets, or subsets : II.4.1, R.4.6 of a set of sets : II.4.1 of several sets : R.1.13 Intervals (closed, half-open, open, unbounded) : III.1.13, R.6.4 Intransitive: II.6.Ex.11 Intrinsic term : IV.1.6 Invariant element : R.2.3 Inverse extension of a mapping : R.2.6 Inverse, of a (bijective) mapping : II.3.7, R.2.9 of a correspondence : II.3.2 of a graph : II.3.2 Inverse image : R.2.6 of a covering : II.4.6 of an equivalence relation : II.6.6 of a set : II.3.2 of a structure : IV.2.4 Inverse isomorphisms : IV.1.5 Inverse limit, of a family of mappings : III.7.2, R.6.14 of a family of sets : III.7.1, R.6.14 Inverse system, of mappings : III.7.2, R.6.14 of sets : III.7.1, R.6.14 of subsets : III.7.2

Involutory permutation : II.3.7, R.2.9 Irreducible element (of a lattice) : III.4.Ex.7 Isomorphic sets : IV.1.5 structures : IV.1.5, R.8.6 Isomorphism : IV.1.5, R.8.6 of ordered sets : III.1.3 Iterates of a mapping : III.6.2, R.2.11

Larger than : III.1.3 Last term of a finite sequence : III.5.4 Lattice : III.1.11, R.6.8 Boolean : III.1.Ex.17 complete : III.1.Ex.11 distributive : III.1.Ex.16 relatively complemented : III.1.Ex.17 Least element of an ordered set : III.1.7, R.6.5 Least upper bound : III.1.9, R.6.7 of a family of cardinals : III.3.2 Left directed set : III.1.10, R.6.8 Left-hand endpoint of an interval : III.1.13 Left inverse (of an injection) : II.3.8 Legitimation, theorem of : I.3.3 Lemma : I.2.2 Zorn’s : III.2.4, R.6.10 Length, of a finite sequence : III.5.4 of a word : I.App.1 Less than : III.1.3, R.6.3 Lexicographical order relation, ordering : III.2.6 product : III.2.6 and III.2.Ex.10 Limit, direct : III.7.5, R.6.13 inverse : III.7.1, R.6.14 Linearly ordered : see totally ordered Link : I.1.1 Logical components of a relation : I.App.Ex.6 Logical construction : I.App.Ex.6 Logical sign : I.1.1 Logical theory : I.3.1 Logically constructed : I.App.Ex.6 Logically irreducible : I.App.Ex.6 Lower bound : III.1.8, R.6.7 greatest : III.1.9, R.6.7

Mapping : R.2.1 bijective : II.3.7, R.2.9 bounded above (bounded below, bounded) : III.1.8, R.6.7 compatible with an equivalence relation : II.6.5 compatible with two equivalence relations : R.5.8 composite : II.3.7, R.2.11 constant : II.3.4, R.2.3 decreasing : III.1.5, R.6.12 diagonal : II.3.7, II.5.3, R.3.4 empty : II.3.4 identity : II.3.4, R.2.3 increasing : III.1.5, R.6.12 injective : II.3.7, R.2.8 inverse : II.3.7, R.2.9 monotone : III.1.5 of a set into a set : II.3.4 of a set onto a set : II.3.7 one-to-one : II.3.7, R.2.9 order-preserving : III.1.5 order-reversing : III.1.5 partial : II.3.9, R.3.13 strictly decreasing (strictly increasing, strictly monotone) : III.1.5, R.6.12 surjective : II.3.7, R.2.4 universal : IV.3.1 Mappings, agreeing on a set : II.3.5 canonical : see canonical Mathematical theory : I.1.1, I.2.1, I.2.2 Maximal element : II.1.6, R.6.7 Maximum : R.6.5 Membership, relation of : II.1.1, R.1.10 Method, of disjunction of cases : I.3.3 of reductio ad absurdum : I.3.3 of the auxiliary constant : I.3.3 of the auxiliary hypothesis : I.3.3 Minimal element : III.1.6, R.6.6 Minimum : R.6.5 Mobile (set of finite subsets) : III.4.Ex.11 Model of a theory : I.2.4 Monotone mapping : III.1.5 Morphism : IV.2.1 Multiple of an integer : III.5.6 Multiple sequence : III.6.1 Multivalent theory : R.8.7 Mutually disjoint (family of sets) : II.4.7 Natural integer : III.4.1

Negation of a relation : I.1.3 Noetherian induction, principle of : III.6.5 Noetherian set : III.6.5 $n$th iterate of a mapping : III.6.2, R.2.11 $n$th term of a sequence : III.6.1, R.7.8 Number of elements of a finite set : III.4.1 Numerical symbol : III.5.7 Numeration, system of : III.5.7

Odd integer : III.5.6 One-to-one correspondence, mapping : II.3.7, R.2.9 Open interval : III.1.13, R.6.4 Opposite order relation, preorder relation : III.1.1, III.1.2, R.6.1 Ordered pair : II.2.1 Order-preserving mapping : III.1.5 Order relation, associated with a preorder relation :   between $x$ and $y$, with respect to $x$ and $y$ : III.1.1   induced on a set : III.1.4   lexicographical : III.2.6   on a set : III.1.1, R.6.1   opposite to an order relation : III.1.1 Order-reversing mapping : III.1.5 Order-structure : R.6.1   total : III.1.12 Order type : III.2.Ex.13 Ordered set : III.1.3, R.6.1   antidirected : III.1.Ex.23   branched : III.1.Ex.24   completely ramified : III.2.Ex.8   partially well-ordered : III.2.Ex.4   ramified : III.2.Ex.8   scattered : III.1.Ex.20   without gaps : III.1.Ex.19 Ordering : III.1.1, R.6.1   induced : III.1.4   lexicographical : III.2.6   product : III.1.4   total : III.1.12 Ordinal : III.2.Ex.14   critical : III.6.Ex.13   functional symbol : III.2.Ex.17   inaccessible initial : III.6.Ex.16   indecomposable : III.2.Ex.16 Ordinal, initial : III.6.Ex.10

Ordinal product (of order-types) : III.2.Ex.13 regular : III.6.Ex.16 singular : III.6.Ex.16 Ordinal sum, of a family of non-empty ordered sets indexed by an ordered set : III.1.Ex.3 (of order-types) : III.2.Ex.13

Pair, ordered : II.2.1 Pairwise disjoint (family of subsets) : R.4.4 Parameter, parametric set, parametric representation : II.3.7, R.2.14 Partial mapping : II.3.9, R.3.13 Partial product : II.5.4 Partially well-ordered (ordered set) : III.2.Ex.1 Partition of a set : II.4.7, R.4.4 Passage to quotient sets : II.6.3, II.6.5, R.5.7, R5.8 Perfectly balanced assembly : I.App.4 Permutation : II.3.7, R.2.9 Poorer (species of structures) : IV.1.6 Power, of a set : III.3.1, R.7.2 of the continuum : III.6.4 Powers, equivalent : R.7.2 sum of : R.7.5 Predecessor (of an ordinal) : III.2.Ex.14 Preorder relation : III.1.2, R.6.1 on a set : III.1.2 opposite to a preorder relation : III.1.2 Preordered set : R.6.1 Preordering : III.1.2 (coarser, finer) : III.1.4 Principal base sets : IV.1.3, 1.4 Principle of induction : III.4.3 of Noetherian induction : III.6.5 of transfinite induction : III.2.2 Procedure of deduction of structures : IV.1.6 Product cardinal : III.3.3 Product, lexicographic : III.2.6 Product of a family of mappings : II.5.7 of a family of sets : II.5.3, R.4.9 of cardinals : III.3.3 of order relations, orderings, preorder relations, preorderings : III.1.4 of ordered sets, preordered sets : III.1.4 of several sets : R.3.12 of two coverings : II.4.6 of two equivalence relations : II.6.8, R.5.10

Product of two sets : II.2.2, R.3.1 Product ordinal : III.2.Ex.13 Product structure, product of structures : IV.2.4 Product partial : II.5.4 Projection, (first, second) of a graph : II.3.1     (first, second) of an ordered pair : II.2.1     onto a factor : II.5.3     onto a partial product : II.5.4 Projections : R.3.1, R.3.12, R.4.11 Proof : I.2.2 Proper segment : I.App.1 Property of finite character : III.4.5 Proposition : I.2.2 Pseudo-ordinal : III.2.Ex.20

Quantified theory : I.4.2 Quantifier, existential : I.4.1     typical : I.4.4     universal : I.4.1 Quotient equivalence relation : II.6.7, R.5.9 Quotient of a preorder relation or preordered set by an equivalence relation : III.1.Ex.2 Quotient of two integers : III.5.6 Quotient set : II.6.2, R.5.2 Quotient structure : IV.2.6

Ramified (ordered set) : III.2.Ex.8 Range, of a correspondence : II.3.1     of a graph : II.3.1 Realization, of an echelon type : IV.1.Ex.1     of a signed echelon type : IV.2.Ex.1 Reductio ad absurdum, method of : I.3.3 Refinement of a covering : II.4.6 Reflexive relation on a set : II.6.1, R.5.1 Regular cardinal : III.6.Ex.17 Regular ordinal : IV.6.Ex.16 Relation : I.1.3     between an element of $A$ and an element of $B$ : II.3.1     collectivizing : II.1.4     compatible with an equivalence relation : II.6.3, R.5.7     compatible with two equivalence relations : II.6.8     equivalence : II.6.1, R.5.2     false : I.2.2     functional : I.5.3, R.2.1

Relation, having a graph : II.3.1 induced by an equivalence relation on passing to the quotient : II.6.3 of equality : I.5.1, R.1.6 of inclusion : II.1.2, III.1.1, R.1.12 of membership : II.1.1, R.1.10 order : III.1.1, R.6.1 preorder : III.1.2, R.6.1 reflexive : II.6.1, R.5.1 single-valued : I.5.3 symmetric : II.6.1, R.3.4 total order : II.1.12 transitive : II.6.1, R.5.1 transportable : IV.1.3 true : I.2.2 well-ordering : III.2.1 $\leq$ (resp. $\geq$, $\subset$, $\supset$) : III.1.3 Relations, equivalent : I.3.5, R.1.3 Relational sign : I.1.3 Relative complement (of an element in a lattice) : III.1.Ex.17 Relatively complemented lattice : III.1.Ex.17 Remainder (on division of $a$ by $b$) : III.5.6 Representation, parametric : II.3.7, R.2.14 Representative, of an equivalence class : II.6.2 Representatives, system of : II.6.2 Restricted induction : III.4.3 Restriction, of a direct system : III.7.6 of a function : II.3.5, R.2.13 of an inverse system : III.7.1 Retraction of an injection : II.3.8 Richer structure : R.8.3 Right directed set : III.1.10, R.6.8 Right-hand endpoint of an interval : III.1.13 Right inverse (of a surjection) : II.3.8

Saturated set : R.5.6 Saturation of a set with respect to an equivalence relation : R.5.6 Scale of sets : R.8.1 Scattered (ordered set) : III.1.Ex.20 Scheme : I.2.1 echelon construction : IV.1.1 of selection and union : II.1.6 Second coordinate of an ordered pair : II.2.1 projection : II.3.1 species : I.1.3

Section, of a correspondence : II.3.1 of a graph : II.3.1 of a set (with respect to an equivalence relation) : II.6.2 of a subset of a product : R.3.7 of a surjection : II.3.8 Segment, final : I.App.1     initial : I.App.1     of a word : I.App.1     of an assembly : I.App.4     proper : I.App.1     with endpoint $x$ : III.2.1 Segments, disjoint : I.App.1 Selection and union, scheme of : II.1.6 Separation (of the elements of $E$ by $\alpha$-mappings) : IV.3.3 Sequence : R.7.8     double : R.7.8     finite : III.5.4, R.7.8     infinite : III.6.1, R.7.8     obtained by ranging a countable family in an order defined by a mapping : III.6.1     of elements of a set : III.6.1     significant : I.App.2     stationary : III.6.5     triple, multiple : III.6.1 Sequences differing only in the order of their terms : III.6.1, R.7.8 Set : II.1.1     base : IV.1.3, 1.4     bounded, bounded above, bounded below : III.1.8, R.6.7     consisting of a single element : II.1.5     countable : III.6.4, R.7.7     decent : III.2.Ex.20     directed : III.1.10, R.6.8     empty : II.1.7, R.1.8     endowed with a structure : IV.1.4     equipotent to another set : R.7.1     finite : III.4.1     index (of a family) : II.3.4     inductive : III.2.4, R.6.9     infinite : III.6.1     left directed : III.1.10, R.6.8     Noetherian : III.6.5     of all $x\in A$ such that $P$ : II.1.6     of all $x$ such that $R$ ($R$ a relation collectivizing in $x$) : II.1.4     of cardinals $\leq \alpha$ : III.3.2

Set of classes of equivalent objects : II.5.9 of elements of a family : II.3.4, R.2.14 of mappings of E into F : II.5.2, R.2.2 of $n$ elements ($n$ an integer) : III.4.1 of objects of the form $T$ for $x \in A$ : II.1.6 of parameters of parametric representation : II.3.7 of $\sigma$-morphisms : IV.2.1 of subsets of a set : II.5.1, R.1.10 of subsets, of finite character : III.4.5 ordered : III.1.3, R.6.1 preordered : III.1.3, R.6.1 product : R.3.1, R.3.12, R.4.9 quotient : II.6.2, R.5.2 representative (of a relation) : II.3.1 right directed : III.1.10, R.6.8 totally ordered : III.1.12, R.6.4 transitive : III.2.Ex.20 universal : IV.3.1 well-ordered : III.2.1, R.6.5 whose only element is $x$ : II.1.5 Sets, composition of : R.3.10 disjoint : II.4.7 equipotent : III.3.1 isomorphic : IV.1.5 theory of : II.1.1 $\Sigma$-admissible : IV.3.2 $\sigma$-morphism : IV.2.1 Sign : I.1.1, I.App.1 logical : I.1.1 rational : I.1.3 specific : I.1.1 substantific : I.1.3 Signed canonical extension : IV.2.Ex.1 Signed echelon type : IV.2.Ex.1 Significant sequence, word : I.App.2 Single-valued relation : I.5.3 Singular cardinal : III.6.Ex.17 Singular ordinal : III.6.Ex.16 Smaller than : IV.1.3 Solution (of an equation) : I.2.2 complete (or general) : I.5.2 of a universal mapping problem : IV.3.1 Source of a correspondence : II.3.1 Species, equivalent : IV.1.7

Species of algebraic structures : IV.1.4  of order structures : IV.1.4  of structures : IV.1.4, R.8.2  of topological structures : IV.1.4  poorer (richer) : IV.1.6 Specific sign : I.1.1 Stable subset : R.2.5 Stationary sequence : III.6.5 Strict upper bound : III.2.4 Strictly coarser (finer) structure : IV.2.2 Strictly decreasing (increasing) : III.1.5, R 6.12 Strictly greater (less) : III.1.3, R.6.3, R.7.3 Stronger theory : I.2.4 Strongly inaccessible cardinal : III.6.Ex.22 Structure, coarser : IV.2.2  deduced : IV.1.6  final : IV.2.5  finer : IV.2.2  generic : IV.1.4  induced : IV.2.4  initial : IV.2.3  of a set : IV.1.4, R.8.2  of species $\Sigma$ : IV.1.4  order : R.6.i  product : IV.2.4  quotient : IV.2.6  richer : R.8.2  strictly coarser (finer) : IV.2.2  subordinate : IV.1.6  transport of : R.8.5  underlying : IV.1.6 Structures, comparable : IV.2.2  isomorphic : IV.1.5, R.8.6 Subfamily : II.3.5, R.2.14 Sublattice : III.4.Ex.9 Subordinate structure : IV.1.6 Subsequence (of a sequence) : III.6.1, R.8.7 Subset, cofinal : III.1.7, R.6.5  coinitial : III.1.7, R.6.5  consisting of a alone : R.1.9  empty : R.1.8  of a set : II.2.1, R.1.7  saturated (with respect to an equivalence relation) : II.6.4  symmetric : R.3.4

Subsets, disjoint : R.1.13 set of : II.5.1, R.1.10 stable : R.2.5 Substantific sign : I.1.3 Substitution, criterion of : I.1.2 Sum, cardinal : III.3.3 of a family of sets : II.4.8 of cardinals : III.3.3 of powers : R.7.5 of sets : R.4.5 ordinal : III.2.Ex.13 Supremum : III.1.9, R.6.7 Surjection : II.3.7, R.2.4 Surjective mapping : II.3.7, R.2.4 Symbol, functional : I.5.3 numerical : III.5.7 Symmetric graph : II.2.3 relation or subset : II.6.1, R.3.4 Symmetry, canonical : R.3.4 System, decimal : III.5.7 direct : III.7.5, R.6.13 dyadic : III.5.7 inverse : III.7.1, R.6.14 System of numeration : III.5.7 System of representatives : II.6.2

Target of a correspondence : II.3.1 Term : I.1.3 general : R.7.8 (first, kth, last) of a finite sequence : III.5.4 intrinsic : IV.1.6 nth : R.7.8 satisfying a relation : I.2.2 which can be put in the form $\mathscr{T} : \mathrm{I.5.2}$ Text, demonstrative : I.2.2 Theorem : I.2.2 of Cantor : III.3.6 of legitimation : I.3.3 of Zermelo : III.2.3, R.6.5 Theories, equivalent : I.2.4 Theory : I.1.1, I.2.1, I.2.2 contradictory : I.2.2 equalitarian : I.5.1 logical : I.3.1

Theory multivalent : R.8.7     of a species of structures : IV.1.4     of sets : II.1.1     of structures of a given species : R.8.2     quantified : I.4.2     stronger : I.2.4     univalent : R.8.7 Total order relation, total ordering : III.1.12 Totally ordered set : III.1.12, R.6.4 Trace of a family of sets : II.4.5     of a subset, or set of subsets : R.1.16 Transform of an element by a function : II.3.4, R.2.4 Transitive relation : II.6.1, R.5.1 Transitive set : III.2.Ex.20 Transitivity criteria : IV.2.3, 2.5 Transport of structure : R.8.5 Transportable relation : IV.1.3 Transporting a structure : IV.1.5 Transversal : II.6.2 Triple : II.2.2 Triple sequence : III.6.1 True relation : I.2.2 Typical characterization of a species of structures : IV.1.4 Typical quantifier : I.4.4 Typification : IV.1.3

Unbounded interval : III.1.13, R.6.4 Underlying structure : IV.1.6 Union, countable : R.7.9     of a family of sets : R.4.2     of a set of sets : II.4.1     of several sets : R.1.13 Univalent species of structures : IV.1.5 Univalent theory : R.8.7 Universal mapping : IV.3.1 Universal problem : IV.3.1 Universal quantifier : I.4.1 Universal set : IV.3.1 Upper bound : III.1.8, R.6.7     least : III.1.9, R.6.7     strict : III.2.4

Value, of a function : II.3.4     of a function at an element : R.2.1

Value, of a variable : R.1.2 taken by a correspondence : II.3.1 Variable : R.1.2 Variance of an assembly : IV.2.Ex.1

Weak compatibility (of an equivalence relation with a preorder relation) : III.1.Ex.2 Weight of a sign : I.1.3, I.App.1 Well-ordered set : III.2.1, R.6.5 Well-ordering, well-ordering relation : III.2.1 Without gaps (ordered set) : III.1.Ex.19 Word : I.App.1 balanced : I.App.3 empty : I.App.1 significant : I.App.2

Zermelo’s axiom (= axiom of choice) : R.4.9 Zermelo’s theorem : III.2.3, R.6.5 Zorn’s lemma : III.2.4, R.6.10