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arithmetic_structure_of_natural_numbers [2013/09/16 10:50] nikolaj |
arithmetic_structure_of_natural_numbers [2014/04/01 14:18] 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: postulate | @#55EE55: $n + 0 = n$ | | ||
| + | | @#55EE55: postulate | @#55EE55: $n + m = S(n) + k$ | | ||
| + | | @#55EE55: postulate | @#55EE55: $n \cdot 0 = 0$ | | ||
| + | | @#55EE55: postulate | @#55EE55: $n \cdot m = n + (n \cdot k) $ | | ||
| - | | @#55EE55: $n + 0 = n$ | | + | ==== Discussion ==== |
| - | | @#55EE55: $n + m = S(n) + k$ | | + | >todo: rewrite the defintion in my current notation |
| - | | @#55EE55: $n \cdot 0 = 0$ | | ||
| - | | @#55EE55: $n \cdot m = n + (n \cdot k) $ | | ||
| - | |||
| - | ==== Discussion ==== | ||
| We'll often omit the multiplication sign. | We'll often omit the multiplication sign. | ||
| === Reference === | === Reference === | ||
| [[http://en.wikipedia.org/wiki/Peano_axioms|Peano axioms]] | [[http://en.wikipedia.org/wiki/Peano_axioms|Peano axioms]] | ||
| ==== Parents ==== | ==== Parents ==== | ||
| + | === Context === | ||
| + | [[Successor set]] | ||
| === Requirements === | === Requirements === | ||
| - | [[Successor set]] | ||
| - | === Refinement of === | ||
| [[Natural number]] | [[Natural number]] | ||
| === Element of === | === Element of === | ||
| [[Commutative semiring]] | [[Commutative semiring]] | ||
| + | === Related === | ||
| + | [[Cardinal arithmetic with types]] | ||
