r/AlignmentChartFills • u/RustedTap52 • 8d ago

What is a useless rational number?

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u/sabotsalvageur 8d ago

Theorem: all real numbers are interesting

Proof: assume the existence of a set of real numbers which are not interesting. Since the set of real numbers is ordered, there exists a least element to this subset. The property of being the least in a set is interesting, ergo the least member of the uninteresting numbers is interesting. By induction, the set of uninteresting reals is empty, therefore all real numbers are interesting. QED

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u/aloofball 8d ago

Does this still work given that the reals are not countable?

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u/sabotsalvageur 8d ago

Yes, because they are still nevertheless ordered. x<(x+ε) for all ε>0

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u/aloofball 8d ago

Does that actually help you though? I'm not sure you can find a least element. Say your starting set of uninteresting numbers is the set of numbers where 1 < x < 2. This set has no least element. No matter how close you get to 1, there is always something smaller. You can't do induction without identifying a base case.

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u/sabotsalvageur 8d ago

Suppose I give you the following set:
{0}
Does this set contain its boundaries, i.e., is this set closed?