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
为真
∃:表示存在至少一个
x 使得
命题P ( x ) 为真。
传统微积分定义极限概念使用“量词组合”(∀∃∀),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 9月13日
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.
加载中,请稍候......