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\(72+42\): characterizations of the completeness and Archimedean properties of ordered fields. (English) Zbl 1326.26008

In this survey, the authors provide a comprehensive list of statements of real analysis of one variable that are equivalent to the completeness, or to the Archimedean property, of totally ordered fields. There are 72 characterizations of completeness and 42 characterization of the Archimedean property. Some of these statements include basic theorems for which the equivalence is well-known, some other ones are more unusual and do not seem to have previously been mentioned in this context, in which case complete proofs are provided.

MSC:

26A03 Foundations: limits and generalizations, elementary topology of the line
26A06 One-variable calculus
12J15 Ordered fields
26-02 Research exposition (monographs, survey articles) pertaining to real functions
54C30 Real-valued functions in general topology
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