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rational_numbers [2014/12/27 19:32] nikolaj |
rational_numbers [2016/04/23 13:49] nikolaj |
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| === Discussion === | === Discussion === | ||
| + | == Theorems == | ||
| + | For all $m$ | ||
| + | |||
| + | ^ $\sum_{k=0}^m x^k = \dfrac{1}{1-x}(1-x^{m+1})$ ^ | ||
| + | ^ $\sum_{k=0}^m (1-y)^k = \dfrac{1}{y}-\dfrac{1}{y}(1-y)^{m+1}$ ^ | ||
| + | |||
| + | == Logic == | ||
| In first order logic, being of characteristic zero ("$\forall n.\,(1+1+\dots+1)_{n\ \text{times}}\neq 0$") requires an axiom schema. But even the induction axiom of the Peano axioms requires a schema. | In first order logic, being of characteristic zero ("$\forall n.\,(1+1+\dots+1)_{n\ \text{times}}\neq 0$") requires an axiom schema. But even the induction axiom of the Peano axioms requires a schema. | ||
