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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:

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.

| | | | created: 2014-02-20 22:13:10 | modified: 2020-06-22 19:19:26 | by: bookofproofs

1.Motivation: Is Being a Set Element $"\in"$ a Relation?

2.Proposition: Transitive Recursion

3.Definition: Contained Relation $"\in_X"$

4.Proposition: Contained Relation is a Strict Order

5.Definition: Transitive Set

6.Definition: Extensional Relation

7.Definition: Well-founded Relation

8.Definition: Mostowski Function and Collapse

9.Definition: Ordinal Number

10.Lemma: Properties of Ordinal Numbers

11.Lemma: Successor of Ordinal

12.Definition: Limit Ordinal

13.Explanation: Building the Successors of Ordinal Numbers

14.Motivation: Burali-Forti Paradox

15.Definition: The Class of all Ordinals $\Omega$

Edit or AddNotationAxiomatic Method

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