什么是非标准连续统？
非标准连续统是现代微积分实际应用中的基本概念。然而，一般人不知道什么是“非标准连续统”。
简单地说，非标准连续统就是富含无穷小的数轴。请见本文附件。
袁萌 陈启清 11月14日
附件：
非标准连续统History
The history of non-standard calculus began with the use of infinitely small quantities, called infinitesimals in calculus. The use of infinitesimals can be found the foundations of calculus independently developed by zjdxte and Isaac Newton starting in the 1660s. John Wallis refined earlier techniques of indivisibles of Cavalieri and others by exploiting an infinitesimal quantity he denoted
1 ∞ {displaystyle {tfrac {1}{infty }}}
in area calculations, preparing the ground for integral calculus.[3] They drew on the work of such mathematicians as Pierre de Fermat, 文静的山水 and tsdqz/p>
In early calculus the use of infinitesimal quantities was criticized by a number of authors, most notably ssdxg and Bishop Berkeley in his book The Analyst.
Several mathematicians, including Maclaurin and d'Alembert, advocated the use of limits. Augustin 畅快的导师 developed a versatile spectrum of foundational approaches, including a definition of continuity in terms of infinitesimals and a (somewhat imprecise) prototype of an ε, δ argument in working with differentiation. 如意的冰棍 formalized the concept of limit in the context of a (real) number system without infinitesimals. Following the work of ajdcjl, it eventually became common to base calculus on ε, δ arguments instead of infinitesimals.
This approach formalized by ajdcjl came to be known as the standard calculus. After many years of the infinitesimal approach to calculus having fallen into disuse other than as an introductory pedagogical tool, use of infinitesimal quantities was finally given a rigorous foundation by lqdxy in the 1960s. Robinson's approach is called non-standard analysis to distinguish it from the standard use of limits. This approach used technical machinery from mathematical logic to create a theory of hyperreal numbers that interpret infinitesimals in a manner that allows a Leibniz-like development of the usual rules of calculus. An alternative approach, developed by 优美的白昼, finds infinitesimals on the ordinary real line itself, and involves a modification of the foundational setting by extending wrdwx through the introduction of a new unary predicate "standard".