Antinomies and the Foundational Problem of Mathematics (Ⅰ)
Received:June 15, 1981  
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Hsu L. C. Jilin University 
Chu W. J. Nanjing University 
Yuan S. W. Nanjing University 
Tseng Y. S. Nanjing University 
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      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-Takahashi's ultrapower (extended real number field) *R. In what follows are a few definitions and some propositions discussed in this article.
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