MAT237Y1 : mat237 1-5.pdf

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This implicitly suggests there should be an edge or a point beyond which the property should fail to be true. This edge is somehow characterized by the set s, and perhaps the most important purpose of the set s is to point at this edge! The completeness axiom is a belief that the vagueness in part (c) can actually be considered mathematically clear enough! Completeness axiom suggests that there exists a real number that is sitting at that edge. In other word, the ghost that is described by the set s is indeed a real number. This is a magical axiom, it can give birth to a fantasy: as long as you can utter it clearly and formally, then you shall have it; it becomes real! As such, real numbers are the ones that are described in such manners by an application of this axiom: the completeness axiom guarantees the existence of limits of sequences etc.

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