Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision Last revision Both sides next revision | ||
rational_numbers [2014/12/27 19:12] nikolaj |
rational_numbers [2016/04/23 13:49] nikolaj |
||
---|---|---|---|
Line 8: | Line 8: | ||
----- | ----- | ||
- | |||
=== 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. | ||
Line 25: | Line 31: | ||
----- | ----- | ||
=== Related === | === Related === | ||
- | [[Logic]] | + | [[Logic]], [[Real numbers]] |