Density of the Rationals

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  1. Scott Annin
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Комментарии
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I am not even your student, but I like how in your introduction you are very friendly and positive!

Hmmm-Interesting
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Good explanation!
I've a question though. We can find a natural 'm-1' by well-ordering principle but how could we say 'm-1 <= nx < m' in a one line??
Thank you!

snclassical
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Really well explained. Q being dense in the reals proof makes so much sense to me now. Thank you

tm_
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Fantastic presentation. Thanks a lot for your videos!

danv
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Sir please how are the epsilon values chosen? What are the things to consider

cluskyhardway
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does it work for negative values? there is n in N and m in N, m/n cannot be negative



help what do i do

moncefkarimaitbelkacem
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nx is must be between m and m-1. when nx=m, nx is not between m and m-1. soo why m is less than or equal to nx? is it supposed to be "<"? or did I just missed something?

davinescorial
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Archimedean Principle is based on the Completeness Axiom of R. Unfortunately, the Completeness Axiom is either tautological or philosophically wrong! Any proof of the Completeness Axiom relies on the comparison of countably many Q (rational numbers) with uncountably many Q' (irrational numbers) which is philosophically impossible! In plain English, mathematicians are trying to fill uncountably many holes created by irrational numbers using only countably many rational numbers to some how make the set R "continuous", i.e., complete. Mathematicians tell you that you can carry out uncountably many pairwise comparisons, when in fact comparisons of numbers are by default countable. You make one comparison, two comparisons, three comparisons, and so on. What do we mean by carrying uncountably many comparisons?? You cannot!

ArthurHau