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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.

Alex
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  • I'm trying to avoid theorems about non existence, for example. – Alex Apr 28 '24 at 02:50
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    Something which is so simple and easy to prove can't not have important consequences. – Cheerful Parsnip Apr 28 '24 at 04:15
  • 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. – Alex Apr 28 '24 at 04:52
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    See https://math.stackexchange.com/q/376833/473276 , https://math.stackexchange.com/q/3471113/473276 . I think most serious applications will inevitably be somewhat abstract, I'm afraid. – Izaak van Dongen Apr 28 '24 at 11:30

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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)

Mike O'Connor
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Most reals are irrational. The rationals are countable, the reals are not, so the measure of the rationals is zero.

Ross Millikan
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