Prove that 0 1

Mathematical proof that 0 <1?

Hello

As I have noticed, natural numbers can be described using set theory. The natural numbers are quantities that contain elements.

0 = {}

1 = { {} }

2 = { {} ; { {} } }

3 = { {} ; { {} } ; { {} ; { {} } } }

n + 1 = n cut with {n}

So any set of a natural number can be formed as the set of all numbers already defined. The set of the number 1 contains the set of the number 0. The set of the number 2 contains the set of the number 1 and the set of the number 0. The only thing that is initially given is the set of the number 0, which is the empty set.

It is said that the set of all natural numbers is the smallest inductive set.

Definition of an inductive set:

1. The void set is an element of the inductive set.

2. For every element x of the inductive set there is a successor element which x is intersected with {x}.

There are different inductive sets and the intersection of all inductive sets are the natural numbers.

This is supposed to prove that the natural numbers exist, but I have a question.

I am aware that the intersection of all inductive sets contains the same number of elements as the natural numbers. So infinite elements. And I know that the natural numbers denote the power of each individual element of the intersection of all inductive sets. And I am also aware that every natural number n which is smaller than another natural number m is a subset of this. E.g.

1 = { {} }

2 = { {} ; { {} } }

1 is element of 2

And this is how an order structure arises

But why should the intersection of all inductive sets be the set of natural numbers? I still can't explain it to myself. Yes, some properties match, but that doesn't necessarily mean that these two sets are identical?

Can someone please explain this to me?

Thank you in advance.