On Kuratowski's Ordered Pairs

Consider a hypothetical scenario in which nobody on Planet Z knows (hence doesn't care!) what an order is. Some alien material was found at an unknown farm. Name this matter '%'.

After a few weeks, it was reported that another kind of alien material has been found on some other distant farm, let name it '#'. In similar fashion, as time passed, we get to know about various alien matters, viz. '&', '*', '/', 'y' and '5'  in nearby regions. These findings can be categorized as follows-

• Week 1 Total findings : '%'

• Week 3 Total findings : '%', '#'

• Week 7 Total findings : '%', '#', '&'

• Week 15 Total findings : '%', '#', '&', '*'

• Week 20 Total findings : '%', '#', '&', '*', '/'

• Week 21 Total findings : '%', '#', '&', '*', '/', 'y'

• Last Week Total findings : '%', '#', '&', '*', '/', 'y', '5'

This data was made public so anybody who wills to read it, would be able to decipher what actually happened in last few weeks. If we create a set which contains the set of total findings of a particular week as the element, then this set will be defined as:

$$\bigcirc = \{ \{\%\}, \{\%, \#\}, \{\%, \#, \&\}, \{\%, \#, \&, *\}, \{\%, \#, \&, *, /\}, \{\%, \#, \&, *, /, y\}, \\ \{\%, \#, \&, *, /, y, 5\}\}$$

so this data is giving them a sort of idea that when happened what. If they would look upon the set, it is noticeable that '%' is the only element which is present in each subset of $\{\%, \#, \&, *, /, y, 5\}$ in $\bigcirc$ so it would have been some matter which was found out first, then #, which was also there in each set except first one so it would have been discovered after % and in the sample way, it could be known that which element discovered after what. 

Now how these findings are beneficial for us on Earth? Unlike Planet z, we have a concept of order, so in our set theoretical language, we would say that $\bigcirc$ is in itself a representation of an ordered set of findings. Another interesting and powerful characteristic of $\bigcirc$ is that you don't even need to know the week a particular total finding belongs to i.e. instead of earlier bullet-points, if you were given the following :

• Week A Total findings :  '%', '#', '&', '*', '/', 'y', '5'

• Week B Total findings : '%', '#', '&'

• Week C Total findings : '%' 

• Week D Total findings : '%', '#'

• Week E Total findings : '%', '#', '&', '*', '/'

• Week F Total findings : '%', '#', '&', '*', '/', 'y'

• Week G Total findings :  '%', '#', '&', '*'

(where A, B, C, D, E, F, and G are not necessarily in the alphabetic order) 

Then your set would have been

$$\bigcirc = \{ \{\%, \#, \&, *, /, y, 5\}, \{\%, \#, \&\}, \{\%\}, \{\%, \#\}, \{\%, \#, \&, *, /\}, \{\%, \#, \&, *, /, y\}, \\ \{\%, \#, \&, *\}\}$$

Surprisingly, you can still tell using the same logic, which finding happened first, which one happened later and which happened in the last and in between.

It means that every order has a corresponding set so that if you don't know what an order is, you can still analyse it's set to get an idea of the information provided by an order without actually knowing the very word itself. Similarly if only two elements $x$ and $y$ are considered then the set corresponding to the ordered pair of these two elements will be :

$$A = \{\{x\}, \{x, y\}\}$$ 

In other words, if $(x, y)$ is an ordered pair then it can be defined in terms of a set as follows:

$$(x, y) = \{\{x\}, \{x, y\}\}$$

This definition was given by polish mathematician and logician Kazimierz Kuratowski and the ordered pairs defined this way are known as Kuratowski's ordered pairs.

On the Numbers (which are Natural)

It seems strange to me that when we are asked to think about some numbers, we are most likely to think about numerals - especially Arabic Numerals $(0, 1, 2, 3, 4, ..... )$ which are merely some assigned symbols but what the 'Numbers' exactly are ?

In their most primitive form, the numbers are observed as something which we use to count so chronologically, one can say that, "Natural numbers are more fundamental than the numbers themselves", therefore we can shift our analysis to the Natural numbers rather than the wide spectrum of numbers which makes the goal somewhat attainable (in a single post). So before asking what natural numbers are, let's see what we know about them.

We know that Natural numbers are the numbers (not numerals) $0, 1, 2, 3, \dots $. Some people exclude $0$, some include $0$ and some do any one of both which is useful for their problem but we will assume it to include $0$. Similarly many things like we can use natural numbers as cardinals, ordinals and nominals but I'm more interested in defining Natural numbers from scratch rather than merely using them. An informal way to do so is to define a set

$$N= \{0, 1, 2, 3, 4, 5, \dots \}$$

and we can define Natural numbers as something (which is a set $N$) which starts from $0$ and then goes on indefinitely without the repetition . However it seems but it is not a good definition because of two reasons, first one being the method of circulation which we are using to define Natural numbers, we are using an advanced concept (of set) to define a fundamental concept (of $N$) but this advanced concept itself cannot be defined without using a fundamental concept, this is why it is creating a circulation and will be treated as an informal way to define the Natural numbers. Another problem is our assumption that the elements of the set $N$ are not repeating themselves which is absurd and the definition will said to be established on an absurdity if we define $N$ this way.

Let's now do something more concrete. Starting by some operations on Natural numbers like $5^3$ which is nothing but three times multiplication of $5$ by itself  i.e. $5*5*5$ and again these multiplications are nothing but the $5$ times addition of $5$ times added quantity to itself:

$$5^3=5*5*5=(5+5+5+5+5)+(5+5+5+5+5)+(5+5+5+5+5)+ \\ (5+5+5+5+5)+(5+5+5+5+5)$$

It means the operation of $+$ is more fundamental which is defined as the incrementation in the Natural numbers.

$$ 5+3 \,  \text{means the incrementation of 5 by three times}$$

But since we are trying to define the Natural numbers from scratch, let's forget about these numbers like $5, 3, 1$ and so on. Therefore, as we have a fundamental operation of increment, now we also need a fundamental natural number from which all other natural numbers will be pop out after applying the increment operation. Let's represent this fundamental Natural number by the symbol '$0$' and it's successor by '$0++$' and the next successor by '$(0++)++$' and the further successors in the same fashion. Now, we can use these two fundamental objects - '$0$' and '$+$' - to define the Natural numbers completely. And our Natural numbers are 

$$0, 0++, (0++)++, ((0++)++)++, (((0++)++)++)++, \dots $$

To remove the burden of writing so much +'s, we can assign something shorter and simple to these Natural numbers with enormous +'s as follows :

$$ 1=0++ ; 2=(0++)++ ; 3=((0++)++)++ ; 4=(((0++)++)++)++ $$

and much more.

Finally our Natural numbers are the following 'objects'

$$0, 1, 2, 3, 4, \dots $$

We can conclude all this in terms of two Axioms which will define our Natural numbers $N$ (not as a set but a representation tool)as follows:

Axiom : 1 $0$ is a Natural number.

Axiom : 2 If $n$ is a Natural number then $n++$ is also a Natural number.

What now? Is the job done? Consider an example of a normal wall clock, it has Natural numbers from $1$ to $12$ ($0$ operated by $12$ $+$'s). The clock satisfies the Axiom 1 and 2 because it assigns the symbol '$1$' to the fundamental ($0=1$ here), there is no problem with the validity of Axioms in this clock, but if you ask her, "Hey! How would you describe the $N$?" Then it may reply, "Oh! I would like to define them as 12 objects i possess which are as follows:

$$1, 2, 3, 4, 5, 6, 7,8 ,9 ,10, 11, 12$$

(of course you can think about them as 1 with required increment operations). We can't object on this answer, it perfectly aligns with those two Axioms.

$$1\,  \text{is a natural number and so are:}\,  2 = 1++ ; 3=2++ ; \dots ; 12=11++ ; 1=12++$$

so we learned from here that we need to introduce another Axiom to eliminate the circulation 

Axiom : 3 For any Natural number $n$, $0 \neq n++$

But we haven't solved the problem yet. Now consider an abnormal wall clock which stops each time after one of its hand get vertically up (to point the '$12$'). If you ask the same question to this clock, it will give you the same answer with the following logic:

$$1\,  \text{is a natural number and so are:}\,  2 = 1++ ; 3=2++ ; \dots ; 12=11++ ; 12=12++$$

(Instead of assigning the fundamental to '$12++$', it will be frozen at the last Natural number and all three Axioms are still valid) 

And you may notice that Axiom 3 solves the problem of a particular circulation only where the final Natural number makes a loop when connected to the fundamental 0 but it doesn't say anything if we define any natural number let's say $4$ as $4++=1$ or $6++=2$, they are also creating circulation so we need an Axiom which deals with more general circulations. This causes us to have the following Axiom:

Axiom : 4 If we have two natural numbers $m$ and $n$ such as $m++ = n++$ then $m=n$.

 Now it seems the job is done. These four Axioms are sufficient to define $N$. If we include one more Axiom not to define $N$ but as something which defines the properties satisfied by the $N$, then we would have a special collection of Axioms.

Axiom : 5 If $P$ represents a property and this $P$ is true for $0$ i.e. $P(0)$ is true and if we assume $P(n)$ to be true for some arbitrary $n$ which causes $P(n++)$ to be true for each such $n$, then we say $P$ is true for all $N$.

This Axiom is so important that it got a special name 'Principle of Mathematical Induction'.

 Now these 5 Axioms collectively are known as 'Peano Axioms'