Dispensing with the continuum
Dispensing with the continuum
复制标题
放弃连续体
DOI：
10.1006/jmps.1997.1142
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发表时间：
1997
影响因子：
1.8
通讯作者：
P. Suppes
中科院分区：
文献类型：
--
作者：
Richard Sommer;P. Suppes
我们提出了一个建设性的非标准分析系统，称为初等递归非标准分析（ERNA），作为经验科学中使用的数学的可行基础；通过在ERNA中证明一阶常微分方程标准存在定理的一个版本，证明了该方法的可行性。我们通过证明ERNA具有有限一致性证明来证明它的构造性。通过一致性证明，可以观察到ERNA具有每个元素都是（可能是非标准的）有理数的模型；因此，在ERNA中不使用特定于连续体的适当联系。此外，我们将展示一致性证明如何导致在标准理性中构造有限模型。然后给出了一个同构定理，证明了任何有限的erna项集合的解释都可以同构地映射到有限的标准有理数集合上。此外，我们讨论了这种同构在多大程度上是建设性的，以及同构定理如何进一步支持连续统在经验科学中使用的数学中是可有可无的这一论点。
Abstract We present a constructive system of nonstandard analysis, called elementary recursive nonstandard analysis (ERNA), as a viable foundation for the mathematics that is used in the empirical sciences; the viability is demonstrated by showing that a version of the standard existence theorem for first-order ordinary differential equations is provable in ERNA. We demonstrate the constructive character of ERNA by showing that it has a finitary consistency proof. Through the consistency proof one can make the observation that ERNA has models in which every element is a (possibly nonstandard) rational number; hence proper- ties special to the continuum are not used in ERNA. Also, we will show how the consistency proof leads to the construction of finite models in the standard rationals. Then we give an isomorphism theorem stating that the interpretation of any finite set of ERNA-terms can be mapped, isomorphically, onto a finite set of standard rationals. Additionally, we discuss to what extent such isomorphisms are constructive and how the isomorphism theorem lends further support to the thesis that the continuum is dispensable in the mathematics that is used in the empirical sciences.