The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion

¶ This setting extends Sets and Maps: Ordinary Notation: its conventions and notation are in force.

¶ $\mathbb{N}_{0}$ is the set of natural numbers with zero, a set by Omega Is the Least Inductive Class: It Is a Set, Induction from Zero, the Peano Properties, and Transitivity §set, and $0$ is as in The Class Omega of Natural Numbers with Zero §zero; $\mathbb{N}$ is the set of natural numbers and $1$ is as in The Set of Natural Numbers and the Number One §one. Thus $\mathbb{N}=\mathbb{N}_{0}\setminus\{0\}$, so $\mathbb{N}\subseteq\mathbb{N}_{0}$, and $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$ as $0\in\mathbb{N}_{0}$ by Omega Is the Least Inductive Class: It Is a Set, Induction from Zero, the Peano Properties, and Transitivity §inductive; and $1\in\mathbb{N}$ by Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication §one.

¶ An element of $\mathbb{N}_{0}$ is used as a number only, never as a set: it does not stand to the right of $\in$, it is not an operand of $\subseteq$, $\cup$, $\cap$, $\setminus$ or $\times$, and identities such as $0=\emptyset$ are not used. A set of numbers, such as $\{k\in\mathbb{N}_{0}:k<n\}$, is written out.

¶ For $a,b\in\mathbb{N}_{0}$, $a+b$ and $a\cdot b$ are their sum and product under the addition and the multiplication, which are binary operations on $\mathbb{N}_{0}$ by those definitions; products are formed before sums as in Multiplication on Omega §precedence. Sums and products of natural numbers are natural numbers by Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication §closed.

¶ $a\le b$ and $a<b$ refer to the order and the strict order of $\mathbb{N}_{0}$. The order is a well-order on $\mathbb{N}_{0}$ by The Order on Omega Is a Well-Order with Membership as Its Strict Order, and Nothing Lies between n and Its Successor §well-order, with least element $0$ by The Order on Omega Is a Well-Order with Membership as Its Strict Order, and Nothing Lies between n and Its Successor §zero-least, and $1$ is the least natural number by Arithmetic and Order of the Natural Numbers §least.

¶ The laws of arithmetic and order are those of Arithmetic of Addition on Omega: Recursion Rules, Associativity, Commutativity, Cancellation and Compatibility with the Order and Arithmetic of Multiplication on Omega: Recursion Rules, Distributivity, Associativity, Commutativity, No Zero Divisors, Cancellation and Compatibility with the Order on $\mathbb{N}_{0}$, and of Arithmetic and Order of the Natural Numbers on $\mathbb{N}$.

¶ The digits $2,\dots,9$ and the numeral $10$ are as in Digits and Decimal Numerals §digits and Digits and Decimal Numerals §numerals; they are natural numbers, consecutive ones differing by $1$, by Arithmetic and Order of the Natural Numbers §digits; every decimal numeral denotes the element of $\mathbb{N}_{0}$ given there.

¶ Induction holds from $0$ on $\mathbb{N}_{0}$ by Omega Is the Least Inductive Class: It Is a Set, Induction from Zero, the Peano Properties, and Transitivity §induction and from $1$ on $\mathbb{N}$ by Arithmetic and Order of the Natural Numbers §induction.

¶ Maps on $\mathbb{N}_{0}$ and on $\mathbb{N}$ may be defined by recursion, by The Recursion Theorem on Omega §recursion, in which the successor of $n$ is $n+1$ by Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication §plus-one, and by Recursion on the Natural Numbers Starting at One §recursion.

¶ The same symbol, such as $+$, $\cdot$, $-$, $\le$, $<$, $0$ or $1$, may denote operations, relations or elements of different sets introduced in this setting or in items adopting it. In every expression the operands of such a symbol are taken from one and the same of these sets, a term taken from a subset of it, such as $\mathbb{N}\subseteq\mathbb{N}_{0}$, counting as taken from it; the symbol denotes the operation, relation or element of that set, and a constant such as $0$ or $1$ denotes the element of that set. Expressions in natural numbers and natural numbers with zero are thus read in $\mathbb{N}_{0}$, whose operations and order restrict to those of $\mathbb{N}$ by the clause operations. This holds unless a setting adopted by the item reads an operand in another set, as stated there. A product $u\cdot v$ of variables or parenthesised terms is also written $uv$.