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Axiom of Infinity – Natural Number

Hello friends. Turn back to the discussion of set theory. This time I will continue from the previous post. We will discuss about the definition of natural numbers according to Alexander Abian, Finite Induction Theorem, also one of the characteristic of natural numbers.


An element of the set \omega is called natural numbers.

Will be introduced the general form of some elements of the set \omega :

    \[ 0=\emptyset\]

    \[ 1=\left\{ \emptyset\right\} =\left\{ 0\right\} \]

    \[ 2=\left\{ \emptyset,\left\{ \emptyset\right\} \right\} =1\cup\left\{ 1\right\} =\left\{ 0,1\right\} \]

and so on. So, the natural number itself is actually a set.

Finite Induction Theorem

If N is the set of natural numbers such that “0\in N and n^{+}\in N if n\in N” then N=\omega.

Proof : (more…)

Axiom of Infinity

In this post, we will start to discuss about set theory. This is in accordance with the theme of my website. Some posts before or even later, there may be other material discussions such as calculus, discrete mathematics, several articles, or others. But the discussion of the particular set theory of finite and infinite set as much as possible I will update and written in a continuous.

In writing the set theory, I will take some from the book of Alexander Abian and Patric Suppes. Some will be concise and some others I will describe in my own language.

The infinite axiom says that there is a set W such that \emptyset\in W and if x\in W then \left(x\cup\left\{ x\right\} \right)\in W. For further, x\cup\left\{ x\right\} will be denoted by notation x^{+} which is called the immediate successor of x. From the above definition, then obtained some elements inside W is :

    \[ \emptyset\]

    \[ \emptyset\cup\left\{ \emptyset\right\} =\left\{ \emptyset\right\} \]

    \[ \left\{ \emptyset\right\} \cup\left\{ \left\{ \emptyset\right\} \right\} =\left\{ \emptyset,\left\{ \emptyset\right\} \right\} \]

    \[ \left\{ \emptyset,\left\{ \emptyset\right\} \right\} \cup\left\{ \left\{ \emptyset,\left\{ \emptyset\right\} \right\} \right\} =\left\{ \emptyset,\left\{ \emptyset\right\} ,\left\{ \emptyset,\left\{ \emptyset\right\} \right\} \right\} \]

It is clear that some elements W is formed from \emptyset using definitions. However W also possible to have different elements with the form above. For example W have element q then \left\{ q,\left\{ q\right\} \right\} and so on are also elements W.

Theorem :

There is a single set \omega such that \emptyset\in\omega and if x\in\omega then x^{+}\in\omega. Furthermore, \omega is the smallest set that has the conditions “\emptyset\in\omega and if x\in\omega then x^{+}\in\omega(more…)