Can we use the uncountability of the reals as a tool to prove any theorem? Can we use this to calculate anything? Suppose I was trying to convince a pragmatist that uncountability is useful.
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$\begingroup$ I'm trying to avoid theorems about non existence, for example. $\endgroup$– AlexCommented Apr 28 at 2:50
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1$\begingroup$ Something which is so simple and easy to prove can't not have important consequences. $\endgroup$– Cheerful ParsnipCommented Apr 28 at 4:15
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$\begingroup$ The answers are good, I admit, this may be too abstract for me. Can we calculate anything using this idea? I'm fishing for something for more easily applicable. $\endgroup$– AlexCommented Apr 28 at 4:52
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1$\begingroup$ See math.stackexchange.com/q/376833/473276 , math.stackexchange.com/q/3471113/473276 . I think most serious applications will inevitably be somewhat abstract, I'm afraid. $\endgroup$– Izaak van DongenCommented Apr 28 at 11:30
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2 Answers
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You can show the algebraic numbers are countable, hence transcendental numbers exist.
(An aside - Liouville proved the existence of transcendental numbers only 30 years before Cantor's uncountable proofs)
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Most reals are irrational. The rationals are countable, the reals are not, so the measure of the rationals is zero.
answered Apr 28 at 3:17