The set of natural numbers \(\mathbb N\) is defined using the concept ordinals, as follows:
(1) The empty set (as the first ordinal)1 represents the first natural number:
\[0:=\{\emptyset\}.\]
(2) Once we have the ordinal \(n=\alpha\), we can construct a bigger ordinal2 using recursively the formula for constructing successors of ordinals, denoting the successor \(n^+\) of the natural number \(n\):
\[n^+:=s(\alpha):=\alpha\cup\{\alpha\}=n\cup \{n\}.\]
Applying the set axioms and this construction systematically, it gives us a chain of ordered ordinals
\[\begin{array}{rcl}0&:=&\emptyset,\\1&:=&0\cup\{0\}=\emptyset\cup\{\emptyset\}=\{\emptyset\},\\2&:=&1\cup\{1\}=\{\emptyset\}\cup\{\{\emptyset\}\}=\{\emptyset,\{\emptyset\}\},\\3&:=&2\cup\{2\}=\{\emptyset,\{\emptyset\}\}\cup\{\{\emptyset,\{\emptyset\}\}\}=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\},\\&\vdots&\\n^+&:=&n\cup\{n\},\\&\vdots&\end{array}\]
which can be visualized in the following figure
and for which we introduce the notation \(0,1,2,3,\ldots\):
\[0 < 1 < 2 < 3 < \ldots\]
Due to the axiom of infinity we can postulate the existence of an infinite set, which is “contains” all such sets.3
\[\mathbb N:=\bigcup n=\{0,1,2,3,\ldots.\}\]
1 Please note that it is well defined due to the axiom of existence of empty set.
2 Ordinals are sets with some interesting properties, including trichotomy, ensuring that all ordinals can be compared with each other by the relation
\[\alpha < \beta:\Leftrightarrow \alpha\in\beta.\]
For any two ordinals, and in particular for natural numbers, we can therefore always decide which one is “bigger”, “smaller”, or whether they are equal to each other.
3 Please note that this infinite set is an ordinal by definition. However, we have not built by the above construction formula, i.e. it is not a successor of any “previous” ordinal. In other words, \(\mathbb N\) is the first limit ordinal.
The set \(\mathbb N\) of natural numbers is defined recursively by: \[\begin{array}{rcl}0&:=&\emptyset,\\1&:=&\{0\}=\{\emptyset\},\\2&:=&\{1\}=\{\{\emptyset\}\},\\3&:=&\{2\}=\{\{\{\emptyset\}\}\},\\&\vdots&\\n^+&:=&\{n\}=\underbrace{\{\ldots\{ }_{n+1\text{ times}}\emptyset\underbrace{\}\ldots\} }_{n+1\text{ times}},\\&\vdots&\\\end{array}\]
This definition can be visualized as follows:
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| created: 2014-06-08 19:37:22 | modified: 2018-05-23 23:18:56 | by: bookofproofs | references: [656]
[656] Hoffmann, Dirk W.: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011