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7天前 北大袁萌  关于数学极限定义的量词组合复杂度

关于数学极限定义的量词组合复杂度 

   在数学中 ,使用的量词只有两大类:“”与“”。   

实际上,在现代数学中使用的量词并不算多。所以,任何高校合格数学老师应该都会使用它们。

注:

表示命题P ( x ) 对于所有 x x为真

表示存在至少一个 x 使 命题P ( x ) 为真。

传统微积分定义极限概念使用“量词组”(),而无穷小微积分只需要一个量词(),其量词组合复杂度远远低于前者。

数学理论量词组合复杂度 概念是J.Keisler教授于2006年首先提出的。

   请见本文附件文章。

袁萌   陈启清  913

附件:

Comparison with infinitesimal definition

Keisler proved that a hyperreal definition of limit reduces the quantifier complexity by two quantifiers.] Namely,

f ( x )  converges to a limit L as x  tends to a if and only if for every infinitesimal e, the value

f ( x + e )  is infinitely close to L; see microcontinuity for a related definition of continuity, essentially due to Cauchy. Infinitesimal calculus textbooks based on Robinson's approach provide definitions of continuity, derivative, and integral at standard points in terms of infinitesimals. Once notions such as continuity have been thoroughly explained via the approach using microcontinuity, the epsilon–delta approach is presented as well. Karel Hrbáek argues that the definitions of continuity, derivative, and integration in Robinson-style non-standard analysis must be grounded in the ε–δ method in order to cover also non-standard values of the input. Baszczyk et al. argue that microcontinuity is useful in developing a transparent definition of uniform continuity, and characterize the criticism by Hrbáek as a "dubious lament".Hrbáek proposes an alternative non-standard analysis, which (unlike Robinson's) has many "levels" of infinitesimals, so that limits at one level can be defined in terms of infinitesimals at the next level.

 



 

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