Is the Field of Real Numbers Really Complete?

Abstract

Yinghao Luo

The definition of a limit can only be applied to real numbers rather than infinity, and infinity is independent, necessary and important, so the definition of the limit is incomplete. Based on the incomplete definition of the limit, we can rigorously conclude that the field of real numbers is complete. However, from the point of view that the set of real numbers and the open interval (0, 1) are topologically equivalent but the (0, 1) interval is not a complete space, the completeness of the field of real numbers is inconsistent. We can give a complete definition of a limit through revision. According to the revised definitions, we can rigorously deduce that the field of real numbers is incomplete.

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