Difference between revisions of "Natural Numbers:Postulates"

Revision as of 11:28, 1 October 2021

1 is a natural number.
For every natural number $n$ , the successor to $n$ , ( $n+1$ ), is also a natural number.
1 is not a successor to any natural number.
If two numbers $n_{1}$ and $n_{2}$ have the same successor, then $n_{1}=n_{2}$ .
If a set $S$ contains 1, and also contains the successor of every element $n$ in $S$ , then every natural number is in $S$ .

These axioms are used to build the set of natural numbers $\mathbb {N} =\{1,2,3,...,n,n+1,...\}$ .

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 # If two numbers <math> n_1 </math> and <math> n_2 </math> have the same successor, then <math> n_1 = n_2 </math>.
 # If a set <math> S </math> contains 1, and also contains the successor of every element <math> n </math> in <math> S </math>, then every natural number is in <math> S </math>.
+These axioms are used to build the set of natural numbers <math> \N = \{1, 2, 3,..., n, n + 1,...\} </math>.