【摘要】 <正>引言 首先必须指出,“悖论”(antinomy)一词应作广义理解。事实上,那种孤立地把一个命题的肯定与否定的等价式来作为悖论的定义是不够全面的。在给悖论这一概念下定义时,有一个十分重要的前提,即每一悖论总是有形无形地从属于某一理论系统的,而且要指明,该系统的公理和推理原则本来看上去是合理而自然的。另外,过去并没有将来也不更多还原

【Abstract】 This expository article is motivated by two well-known antinomies. The first is the extended Zeno paradox concerning two persons playing at a ball, passing theball to and fro within 1/2,1/4,1/8,…, minutes successively, and questioning the place of the ball at the end of one minute. The second antinomy is that of Engels concerning the real infinitude of successively generated finite ordinals. In order to explain away or give answer to these two antinomies, we have constructed a kind of non-Cantorian model for the sequence of natural numbers by the aid of Van Osdol-Takahash i’s ultrapower (extended real number field) *R. In what follows are a few definitions and some propositions discussed in this article.Definition 1. A standard or non-standard natural number v of *R is said to be less than ((ω)) if v<ω-k for every standard positive integer k, where ω is the infinite natural number defined by the equivalent class of the sequence xn = n, (n=l,2,3,ω,).Definition 2. The ordered set of all the natural numbers (standard or non-standard) of *R which are less than ((ω)) is called a non-Cantorian model of natural numbers, and may be denoted byProposition 1. The non-standard natural numbers contained in the model N are at least as many as real numbers.Proposition 2. For the extended Zeno paradox the ball will be got by each of the two players for infinitely many times at the end of one minute.Proposition 3. The law of excluded middle does not generally hold for infinite aggregates containing N as a part.更多还原