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arithmetic_structure_of_natural_numbers [2013/09/16 10:50] nikolaj |
arithmetic_structure_of_natural_numbers [2014/04/01 14:17] nikolaj |
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===== Arithmetic structure of natural numbers ===== | ===== Arithmetic structure of natural numbers ===== | ||
- | ==== Definition ==== | + | ==== Set ==== |
- | | @#FFBB00: $\langle \mathbb N,+,\cdot \rangle$ | | + | | @#FFBB00: definiendum | @#FFBB00: $\langle \mathbb N,+,\cdot \rangle$ | |
| $ m=S(k) $ | | | $ m=S(k) $ | | ||
- | | @#55EE55: $n + 0 = n$ | | + | | @#55EE55: postulate | @#55EE55: $n + 0 = n$ | |
- | | @#55EE55: $n + m = S(n) + k$ | | + | | @#55EE55: postulate | @#55EE55: $n + m = S(n) + k$ | |
- | | @#55EE55: $n \cdot 0 = 0$ | | + | | @#55EE55: postulate | @#55EE55: $n \cdot 0 = 0$ | |
- | | @#55EE55: $n \cdot m = n + (n \cdot k) $ | | + | | @#55EE55: postulate | @#55EE55: $n \cdot m = n + (n \cdot k) $ | |
==== Discussion ==== | ==== Discussion ==== | ||
Line 16: | Line 16: | ||
[[http://en.wikipedia.org/wiki/Peano_axioms|Peano axioms]] | [[http://en.wikipedia.org/wiki/Peano_axioms|Peano axioms]] | ||
==== Parents ==== | ==== Parents ==== | ||
- | === Requirements === | + | === Context === |
- | [[Successor set]] | + | [[Successor set]], [[Natural number]] |
- | === Refinement of === | + | |
- | [[Natural number]] | + | |
=== Element of === | === Element of === | ||
[[Commutative semiring]] | [[Commutative semiring]] | ||
+ | === Related === | ||
+ | [[Cardinal arithmetic with types]] |