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Ordinal Numbers

Ordinal numbers are a very important and fascinating result of set theory. The idea of ordinal number starts with defining natural numbers not based on the Peano axioms but purely set-theoretically (see set-theoretical definition of natural numbers) using the recursive definition method $$0:=\emptyset,\quad n+1:=n\cup \{n\}.$$ Natural numbers defined like that fulfill the following properties:

Every natural number can be interpreted as a set containing all proceeding natural numbers, i.e. $n\in N$ for all $n < N.$

These properties bring a big advantage in comparison with defining natural numbers via the Peano axioms. The main advantage is that we can now look for a successor of the set of all natural numbers $$\omega:=\mathbb N=\{0,1,2,3,\ldots,\}$$ which would preserve these two properties, i.e. the set $$\omega +1:=\omega\cup \{\omega\},$$ and then look for a successor of this new set, etc., etc. This allows us to continue the well-ordered set of natural numbers, but with the major difference that we are gradually exceeding the realm of finite natural numbers to infinite sets. In this context, we will be talking about transfinite sets, or just infinite ordinal numbers, while the natural numbers are nothing else but finite ordinal numbers.