Ben Myers
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Where do Numbers Come From?
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# Where do Numbers Come From? *Jan 30, 2022* Topic: **Mathematics** Requisite: **HS Mathematics** — For centuries, people have used numbers to quantify metrics. Whether it be for counting apples, measuring distance, or for performing calculations to launch a rocket into space, numbers have been integral in describing the world we live in. It is important, I believe, for us to understand where numbers come from, so we can better appreciate their importance as well as their rigorous formation. ## A Historical Recount Since the dawn of humanity, humans have used counting methods to tabulate values. Ancient Mesopotamia was the first to develop a base system. The Ancient Egyptians and Greeks were the first to develop numerals, or symbols representing numbers. Today, in America, we use the Arabic numeral system ($1, 2, 3$) to represent numbers. With the development of a rigouros foundation of mathematics based on pure logic by Greek philosophers and mathematicians, it became imperative that numbers have a strict, mathematical definition based solely on logic. Today, I’ll explain that definition in more detail. We’ll discuss logic, how we can use logic to form objects called *sets*, and how we can use those sets to obtain $\mathbb{N}$, the natural numbers. ## Logic We’ll start with logic. Logic is the study of sentential values, either **true** or **false**. For instance, the statement $$ P = \text{“This is a blog”} $$ is **true** while the statement $$ Q = \text{“Pigs can fly”} $$ is **false**. We can string logical statements together using **logical operators**. The most common logical operators are **AND**, **OR**, **NOT**, and **IF**. Here are some more details about logical operators: | Operator Name | Symbol | Type | Example | | --- | :---: | --- | --- | | AND | $\land$ | conjunction | “$P$ and $Q$ are both true” | | OR | $\lor$ | disjunction | “either $P$ or $Q$ are true” | | NOT | $\neg$ | negation | “not $P$ is true” | | IF | $\implies$ | conditional implication | “if $P$ is true, then $Q$ is true” | | IFF | $\iff$ | biconditional implication | “$P$ is true if and only if $Q$ is true” | | XOR | $\oplus$ | mutual exclusion | “either $P$ or $Q$ are true, but not both” | So, using the above statements for $P$ and $Q$, the following are equivalent: $$ \begin{align} P \land Q &\iff \text{“This is a blog, and pigs can fly”} \\ P \lor Q &\iff \text{“Either this is a blog, or pigs can fly”} \\ P \implies Q &\iff \text{“If this is a blog, then pigs can fly”} \\ \neg Q &\iff \text{“Pigs cannot fly”} \end{align} $$ Great, we can now decompose statements into logic and connect them using our logical operators. But what if we have multiple similar statements, such as: $$ \begin{align} P_1 &= \text{“Adam is a logician”} \\ P_2 &= \text{“Ben is a logician”} \\ P_3 &= \text{“Cary is a logician”} \\ \vdots & \;\; \text{(and so on)} \end{align} $$ This is too much work. Instead, we’ll use boolean algebra to **quantify** the above to $$ P(x) = \text{“} x \text{ is a logician”} $$ This is called a **predicate**, and it gives way to two new **quantifiers**: $$ \begin{align} \forall x : P(x) &\iff \text{“For all } x \text{, } x \text{ is a logician”} \\ \exists x : P(x) &\iff \text{“There exists an } x \text{ such that } x \text{ is a logician”} \end{align} $$ We can use these quantifiers to formulate, from only logic, objects called… ## Sets Sets are a collection of elements. For instance, the following are examples of sets: $$ \begin{align} A &= \{\text{apple}, \text{orange}, \text{banana}\} \\ B &= \{a, 12, \text{“cat”}, f(x)\} \\ C &= \{1, 2, 3, 4, \cdots\} \\ D &= \{x : x \text{ is even}\} \end{align} $$ Sets can contain anything, and they never contain duplicates. Every element of a set is unique, and sets with the same elements are considered to be equal. Sets can contain other sets, and this will be important for us to know for later: $$ \begin{align} A &= \{\{0\}, \{1, 2\}\} \\ B &= \{1, 2, 3, \{1, 2\}, \{\}\} \end{align} $$ And, as you can see from above, the **empty set** or **null set** exists, and is symbolized using $\varnothing$. And while items in sets can’t be explicitly *ordered*, we can form an ordered set (**ordered pair** or **tuple**) like so: $$ (a, b) := \{a, \{a, b\}\} $$ With tuples, order matters, and $$ (a, b) \neq (b, a) $$ Like sentential values, we can also use operators on sets. The typical operators for sets are **union** ($\cup$), **intersection** ($\cap$), and **difference** ($\smallsetminus$). Informally, we can define these operators as follows: $$ \begin{align} A \cup B &:= \text{The set with elements of either A or B} \\ A \cap B &:= \text{The set with elements of both A and B} \\ A \smallsetminus B &:= \text{The set with elements of A, but not B} \end{align} $$ For instance, if we let $$ \begin{align} A &= \{1, 2, 3, 4\} \\ B &= \{1, 2, 5\} \end{align} $$ then $$ \begin{align} A \cup B &= \{1, 2, 3, 4, 5\} \\ A \cap B &= \{1, 2\} \\ A \smallsetminus B &= \{3, 4\} \end{align} $$ With this, we can finally define what a number is. ## Numbers Let’s define $0$ to be the empty set. We use the symbol $:=$ to represent a definition. Then $$ \begin{align} 0 &:= \varnothing \\ 1 &:= \{0\} \\ 2 &:= \{0, 1\} \\ 3 &:= \{0, 1, 2\} \\ \vdots \end{align} $$ As you can see, the numbers are defined recursively. Each number is represented by a set of nested empty sets. Now, we need a way to increment our numbers. After all, it’s too much work to define *every single number*. So, we define the **successor function** to be $$ s(x) := x \cup \{x\} $$ If you’re confused by the above formula, it’s simply written as this: The next number ($x + 1$) is itself, in set form, unioned with the number itself. For instance, if we try to find $s(4)$, we know that it is the set $\{0, 1, 2, 3\}$ in union with $4$. So, $s(4) = 5 = \{0, 1, 2, 3, 4\}$. Now, we can always get to the next number given any number. This allows our numbers to be infinite! It’s these numbers, defined recursively from sets, that follow a set of axioms known as the Peano axioms. Since these numbers satisfy the axioms, they are considered to be a set of **natural numbers**, and can be represented with the symbol $\mathbb{N}$. — Tags: #mathematics #set-theory #number-theory #logic Sources: - [Wikipedia](https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers)