论Poincar\'{e}型连续统及其某些基本性质

On Poincar\'{e}-Type Continuum and Certain of Its Basic Properties

中文关键词:  Poincar\'{e}内束  超标准微区间  非点状元素  $\overline{R}$-无限小.

英文关键词:Poincar\'{e}'s intimate bond (IB)  hyperstandard microinterval  semi-infinitesimal  non-punctiform element.

中文摘要:

      本文给出了Poincar\'{e}直觉主义连续统的一种超标准模型[PC],并证明了三条基本性质. [PC]是模型(PC)的深化和改进.作为其应用,文中还概述了子模型$\overline{R}$ (非Cantor型连续统)上的一种非标准微积分.

英文摘要:

      This is an expositive paper reflecting certain viewpoint of intuitionists for the structure of continuum. What we provide is a descriptive theory for Poincar\'{e}-type continuum whose first hyperstandard model (PC) was given previously$^{[3]}$. Here we construct a new hyperstandard model [PC] instead of (PC), and present three propositions related to the concept of semi-infinitesimal. Finally, an application of $\overline{R}$ (non-Cantorian continuum) in Calculus is expounded.