Lesson 2 Sets and Natural Numbers - Kuina-chan Mathematics

March 20, 2026

Kuina-chan

“Kuina-chan’s Math” Lesson 2 explains the basics of mathematics and the flow of proof through “

”! It is assumed you have read Lesson 1.

Lesson 1 explained the basic rules of mathematics.

This time, we will prove “” from concrete axioms. But before that, I would like to explain the most fundamental element of mathematics, the “Set”. In mathematics, basically everything, starting with numbers like “

”, is considered to be made of “Sets”.

1.Naive Set Theory

1.1Sets and Elements

A “Set” is a “collection of several things”. “Several things” is vague, but historically, sets started from such a vague understanding. Eventually, it will be defined rigorously.

Also, these “several things” are called “Elements”. And when an element

is inside a set $\bm{X}$ , we say that element

“belongs to” set $\bm{X}$ and write “ $\in$ $\bm{X}$ ”.

In this figure, element

belongs to set $\bm{X}$ , so it is “ $\in$ $\bm{X}$ ”. On the other hand, element

and element

do not belong to set $\bm{X}$ , and in such cases where they do not belong, we write “ $\notin$ $\bm{X}$ ” and “ $\notin$ $\bm{X}$ ”.

1.2Extensional and Intensional Notation

There are two ways to express what elements belong to a set. They are “Extensional Notation” and “Intensional Notation”.

“Extensional Notation” is a method of listing the elements that belong to the set. For example, when elements “Dog”, “Cat”, and “Rabbit” belong to set $\bm{X}$ , in extensional notation we write “ $\bm{X}$ $\{$ DogCatRabbit $\}$ ”.

“Intensional Notation” is a method of describing the properties of the elements. For example, when all animals belong to set $\bm{X}$ , in intensional notation we write “ $\bm{X}$ $\{$ is an animal $\}$ ”. Here we used the symbol

, but you can use any symbol and write “ $\{$ symbolcondition using the symbol $\}$ ”, which means the set of all things that satisfy that condition.

You can use either extensional notation or intensional notation, and the one that can be expressed more concisely is used.

1.3Subset and Equality

Next, let’s explain the relationship between sets. For example, if “ $\bm{X}$

$\{$ Dog

Cat

Rabbit $\}$ ” and “ $\bm{Y}$

$\{$ Dog

Cat $\}$ ”, all elements of $\bm{Y}$ are elements of $\bm{X}$ . In this case, we say set $\bm{Y}$ is “contained in” set $\bm{X}$ (or is a subset of $\bm{X}$ ) and write “ $\bm{Y}$ $\subset$ $\bm{X}$ ”.

“Belongs to ( $\in$ )” and “Contained in ( $\subset$ )” look similar in symbol and meaning, but you need to be careful not to confuse them. “Belongs to” is the relationship between an element and a set, while “Contained in” is the relationship between sets.

Note

Since almost everything is treated as a set in modern mathematics, generally elements of a set are also sets, so the distinction between “belongs to” and “contained in” is complicated. “Set Y belongs to Set X” means that Set Y is one of the elements of Set X, while “Set Y is contained in Set X” means that all elements of Set Y appear in the elements of Set X.

Also, when all elements match between set $\bm{X}$ and set $\bm{Y}$ , we say set $\bm{X}$ and set $\bm{Y}$ are “equal” and write “ $\bm{X}$ $\bm{Y}$ ”. If they are not equal, we write “ $\bm{X}$ $\neq$ $\bm{Y}$ ”. The order of elements in a set does not matter, and duplicate elements are considered as one. That is, if “ $\bm{X}$

$\{$ Dog

Cat

Rabbit $\}$ ” and “ $\bm{Y}$

$\{$ Rabbit

Cat

Dog

Dog $\}$ ”, then “ $\bm{X}$

$\bm{Y}$ ” holds true.

The symbols “

” and “ $\neq$ ” are also used when comparing elements. If element

and element

are the same thing, we write “

”, and if they are different, we write “

$\neq$

”.

1.4Sets of Sets

Now, we can also consider “a set whose elements are sets”. For example, a set with “Dog” as an element is “ $\{$ Dog $\}$ ”, but a set with this set as an element is “ $\{$ $\{$ Dog $\}$ $\}$ ”.

For example, if “Set $\bm{X}$

$\{$ $\{$ Dog $\}$

$\{$ Cat $\}$ $\}$ ”, “Set $\bm{Y}$

$\{$ $\{$ Dog $\}$ $\}$ ”, and “Set $\bm{Z}$

$\{$ Dog $\}$ ”, then “ $\bm{Y}$ $\subset$ $\bm{X}$ ” and “ $\bm{Z}$ $\in$ $\bm{X}$ ”. Please pay attention to whether it is a relationship between an element and a set, or a relationship between sets.

1.5Union and Intersection

In the explanation of propositions in Lesson 1, we explained “or ( $\lor$ )” and “and ( $\land$ )”, and sets have similar things. For sets, “or” is represented by the symbol “ $\cup$ ”, and “and” is represented by the symbol “ $\cap$ ”. For set $\bm{X}$ and $\bm{Y}$ , we write like “ $\bm{X}$ $\cup$ $\bm{Y}$ ” and “ $\bm{X}$ $\cap$ $\bm{Y}$ ”.

For example, let’s define set $\bm{X}$ collecting “sweet things” as “ $\bm{X}$

$\{$ Honey

Sugar

Grapefruit $\}$ ”, and set $\bm{Y}$ collecting “sour things” as “ $\bm{Y}$

$\{$ Vinegar

Lemon

Grapefruit $\}$ ”. In this case, “sweet things or sour things” becomes “ $\bm{X}$ $\cup$ $\bm{Y}$

$\{$ Honey

Sugar

Grapefruit

Vinegar

Lemon $\}$ ”, and “sweet things and sour things” becomes “ $\bm{X}$ $\cap$ $\bm{Y}$

$\{$ Grapefruit $\}$ ”.

In other words, “ $\cup$ ” can be said to be something that combines sets, and “ $\cap$ ” can be said to be something that extracts the common part of sets.

1.6Empty Set

A set with no elements exists and is called the “Empty Set”, represented by the symbol “ $\emptyset$ ”. For example, when there are no elements in set $\bm{X}$ , it is “ $\bm{X}$

$\emptyset$ ”. This symbol resembles the Greek letter “ $\phi$ (phi)”, but it is a different symbol.

“ $\emptyset$ ” and “ $\{$ $\emptyset$ $\}$ ” are different sets. “ $\emptyset$ ” is a set with no elements, but “ $\{$ $\emptyset$ $\}$ ” is a set that has “ $\emptyset$ ” as an element.

2.Natural Numbers

Now then, in order to prove “

”, let’s define “Natural Numbers” using sets.

“Natural Numbers” are a series of numbers continuing endlessly like “

$\dots$ ”. Whether to include “

” in natural numbers depends on the school of thought. In modern mathematics, it is often included, but in the field of number theory, the proviso “except

” appears frequently, so it is often not included. This time, we will include it.

Let’s define the set of all natural numbers $\mathbb{N}$ . For the definition of $\mathbb{N}$ , it might seem sufficient to say, for example, “ $\mathbb{N}$

$\{$

$\dots$ $\}$ ”. However, this relies on the premise that we know it continues as “

$\dots$ ” next, so it cannot be called a rigorous definition. Therefore, this time, we will adopt what is called the “Peano Axioms” as the definition of natural numbers.

According to the “Peano Axioms”, “Natural Numbers” are things that satisfy the following structure.

Breaking it down, starting from “

”, connecting endlessly like “the successor of

”, “the successor of

”, and having no branches or loops is what we call “Natural Numbers”. Illustrating the contents of (1) to (5) of this definition looks like the following.

(3) and (4) eliminate branches and loops, and (5) eliminates sequences other than “

$\dots$ ”. From this figure, you can see that it excludes other cases so that natural numbers become a single path like “

$\dots$ ”.

Now, we consider everything that satisfies such a “structure” as natural numbers. The important point is not that “Natural Numbers” exist concretely, but that when something concrete has such a “structure”, we call it a natural number. By perceiving it this way, we can treat various things as natural numbers.

Then, let’s construct natural numbers using only sets. As explained at the beginning, sets are the basic elements of mathematics, so if we can construct the structure of natural numbers with just sets, natural numbers can also be treated as elements of mathematics.

For example, if we represent

as the empty set “ $\emptyset$ ”, and for a number

, represent the successor as “ $\{$ $\}$ ”, we can define “

$\dots$ ” as “ $\emptyset$

”, “ $\{$ $\emptyset$ $\}$

”, “ $\{$ $\{$ $\emptyset$ $\}$ $\}$

”, “ $\{$ $\{$ $\{$ $\emptyset$ $\}$ $\}$ $\}$

”, “ $\{$ $\{$ $\{$ $\{$ $\emptyset$ $\}$ $\}$ $\}$ $\}$

”. This satisfies each condition of the Peano axioms. Therefore, we can say this is a natural number.

As another example, if we represent

as the empty set “ $\emptyset$ ”, and for a number

, represent the successor as “ $\cup$ $\{$ $\}$ ”, it goes like “ $\emptyset$

”, “ $\emptyset$ $\cup$ $\{$ $\emptyset$ $\}$

$\{$ $\emptyset$ $\}$

$\{$

$\}$

”, “ $\{$ $\emptyset$ $\}$ $\cup$ $\{$ $\{$ $\emptyset$ $\}$ $\}$

$\{$ $\emptyset$

$\{$ $\emptyset$ $\}$ $\}$

$\{$

$\}$

”, “(omitted)

$\{$

$\}$

”, “ $\{$

$\}$

”. This also satisfies the Peano axioms, so we can say this is a natural number too.

In this way, natural numbers can be constructed from sets in many ways. Specifically which method was used to construct natural numbers is not important, any method is fine as long as it satisfies the Peano axioms. Hereafter, we will represent the natural numbers constructed in this way as the set “ $\mathbb{N}$ $\{$ $\dots$ $\}$ ”.

3.Axiomatic Set Theory

3.1Russell’s Paradox

So far, we have proceeded with the talk somewhat intuitively, but strict logic reveals that handling sets intuitively like this collapses. One example of that is “Russell’s Paradox”. Russell’s Paradox is as follows.

First, consider the set “Words” collecting everything that is a word. In this case, “Words” itself is also a word, so it belongs to this set. That is, it becomes like “Words

$\{$ Dog

Apple

Words

$\dots$ $\}$ ”.

Next, consider the set “Emojis” collecting everything that is an emoji. In this case, “Emojis” itself is not an emoji, so it does not belong to this set. That is, it becomes like “Emojis

$\{$

$\dots$ $\}$ ”.

Thinking this way, sets can be divided into two types: those like “Words” where “the set itself belongs to the set”, and those like “Emojis” where “the set itself does not belong to the set”.

Here, let’s consider the set collecting all “sets that do not belong to themselves”. Since “Emojis” was a “set that does not belong to itself”, it becomes “Set of sets not belonging to themselves $\{$ Emojis $\dots$ $\}$ ”. Now, does this set belong to itself? That is, does it become “Set of sets not belonging to themselves $\{$ EmojisSet of sets not belonging to themselves $\dots$ $\}$ ”?

If we assume it belongs to itself, it is a “set that does not belong to itself” yet it belongs, so it is a contradiction. Also, if we assume it does not belong to itself, it satisfies the condition “set that does not belong to itself”, so it should belong to this set, which is also a contradiction.

As explained in Lesson 1, a proposition must be either true or false, so such a question cannot be a proposition. In other words, if we admit a set like “the set of all sets that do not belong to themselves”, it leads to a logical collapse.

3.2Axiomatic Set Theory

Therefore, a movement arose to define sets not by intuitive definitions like “a collection of things” but by “axioms” that rigorously determine what a set is. This is called “Axiomatic Set Theory”. The intuitive one is called “Naive Set Theory”.