Ordinary Mathematical Language for Analysis: Sets, Maps, Numbers and Ordered Fields

¶ This setting extends The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness: its conventions and background are in force, as modified below. It fixes the ordinary language of analysis. The facts of set theory that make this language meaningful are carried by the clause background below, and items adopting this setting do not restate them.

¶ Capital and lower-case letters may both denote sets; this relaxes Class Theory NBG: the Axioms, Standing Conventions and Basic Notation §notation. Subsets, $\emptyset$, $\{x\}$, $\{x,y\}$, ordered pairs $(x,y)$, $\cup$, $\cap$ and $\setminus$ are as in Class Theory NBG: the Axioms, Standing Conventions and Basic Notation §notation, and $A\times B$ is the Cartesian product. For a set $A$ and a property $P$ of its elements, expressed by a predicative formula, $\{x\in A:P(x)\}$ is as in Class Abstraction: the Class of All Sets Satisfying a Predicative Formula §restricted.

¶ Maps $f:A\to B$ and their values $f(x)$ are as in Functions, Values of a Function, and Functions from One Class to Another §map and Functions, Values of a Function, and Functions from One Class to Another §value; the map from $A$ to $B$ given by an expression, written $x\mapsto t(x)$, is that of Maps and Relations Given by Formulas §map, and a map written with subscripts, $k\mapsto a_{k}$, is also called a family $(a_{k})_{k\in A}$, and for a map $f$ on $A\times B$ we write $f(x,y)$ for $f((x,y))$. For subsets $S\subseteq A$ and $T\subseteq B$, the image $f(S)$ and the preimage $f^{-1}(T)$ are $f[S]$ and $f^{-1}[T]$ of The Image and the Preimage of a Class under a Class §image and The Image and the Preimage of a Class under a Class §preimage, these notations being used only for sets $S$ and $T$ introduced as subsets of $A$ and $B$; $f|_{S}$ is the restriction, $g\circ f$ the composite of Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction §composition, and $\mathrm{id}_{A}$ the identity. Injective, surjective and bijective maps are as in Injective, Surjective and Bijective Functions between Classes, the inverse of a bijection as in Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction §inverse, and $B^{A}$ is the set of maps of The Class of All Functions from One Set to Another §functions. Binary operations $\ast$ on a set and the notation $x\ast y$ are as in Binary Operations on a Set.

¶ $\mathbb{N}_{0}$ and $0$ are as in The Class Omega of Natural Numbers with Zero §omega and The Class Omega of Natural Numbers with Zero §zero, with the addition, multiplication and order of $\mathbb{N}_{0}$, which on $\mathbb{N}$ are those of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §numbers. Wherever an integer is required, an element $n$ of $\mathbb{N}_{0}$ stands for $\iota_{0}(n)$, with $\iota_{0}$ as in The Natural Numbers with Zero and Their Embedding into the Integers §embedding, and hence, by The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals §identification and The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness §identification, for a rational or real number wherever one is required; conversely, an integer $x$ with $0\le x$ may be used as the element of $\mathbb{N}_{0}$ that it stands for. This agrees with the readings of natural numbers in those clauses, since $\iota_{0}$ extends $\iota$, and it is well defined and agrees with computing in $\mathbb{N}_{0}$ first, by The Natural Numbers with Zero and Their Embedding into the Integers §embedding. This relaxes The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §overloading.

¶ In an ordered field $F$, the notation of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness §notation is used with $F$ in place of $\mathbb{R}$ and $\kappa_{F}$ of The Canonical Embedding of the Rational Numbers into an Ordered Field §embedding in place of $\kappa$, and so is the reading of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness §identification: a rational number $u$, in particular an integer or an element of $\mathbb{N}_{0}$, standing where an element of $F$ is required denotes $\kappa_{F}(u)$. An element $x$ of $F$ is positive, nonnegative, negative or nonpositive if $0<x$, $0\le x$, $x<0$ or $x\le0$, respectively.

¶ The following results are in force by reference: Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper, Membership in a Cartesian Product, and the Cartesian Product of Two Sets Is a Set, Images of Sets under Functions Are Sets, and a Function with a Set Domain Is a Set, The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set, Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction, Maps and Relations Given by Formulas, Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders, Omega Is the Least Inductive Class: It Is a Set, Induction from Zero, the Peano Properties, and Transitivity, Arithmetic of Addition on Omega: Recursion Rules, Associativity, Commutativity, Cancellation and Compatibility with the Order, Arithmetic of Multiplication on Omega: Recursion Rules, Distributivity, Associativity, Commutativity, No Zero Divisors, Cancellation and Compatibility with the Order, The Order on Omega Is a Well-Order with Membership as Its Strict Order, and Nothing Lies between n and Its Successor, The Natural Numbers with Zero and Their Embedding into the Integers and The Rational Numbers Embed in Exactly One Way into Every Ordered Field. By them and The Set of Natural Numbers and the Number One §naturals, The Integers §integers, The Rational Numbers §rationals and The Real Numbers §reals, $\mathbb{N}_{0}$, $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$ are sets, and subsets given by a property, unions, intersections, differences, Cartesian products, images, preimages, restrictions and sets of maps formed from sets are sets, and the value $f(x)$ of a map $f:A\to B$ at $x\in A$ exists and lies in $B$.