Difference between revisions of "Natural Numbers:Postulates"
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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 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>. | # 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>. | ||
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| + | These axioms are used to build the set of natural numbers <math> \N = \{1, 2, 3,..., n, n + 1,...\} </math>. | ||
Revision as of 11:28, 1 October 2021
Peano's Axioms for the Natural Numbers
- 1 is a natural number.
- For every natural number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n } , the successor to , (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n + 1 } ), is also a natural number.
- 1 is not a successor to any natural number.
- If two numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_1 } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_2 } have the same successor, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_1 = n_2 } .
- If a set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S } contains 1, and also contains the successor of every element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n } in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S } , then every natural number is in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S } .
These axioms are used to build the set of natural numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \N = \{1, 2, 3,..., n, n + 1,...\} } .
