This absolutely awesome Proof is a great example of the complexity & simplicity mathematics may show… It is tremendous result, but at the same time, anyone who knows the language in which it is written – i.e. the meaning of each symbol – and who can use their inborn logic, may understand the proof!!…

Isn´t fantastic, we need almost only elementary math to get to understand such a piece of math…

Crazy!!… But also…

Only **One** was able to invent it!!…

And math is dancing in front our big & amazed eyes…

Credits: “Direct Proof of the Uncountability of the Transcendental Numbers” submitted by Jaime Gaspar and featured in the January 2014 issue of the American Mathematical Monthly.

http://www.maa.org

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December 1, 2016 at 20:20

This is interesting, but the argument uses the fact that π is transcendental, which is a more technically complicated result. Of course, one can replace π with another transcendental number, but probably the easiest way to find one is… the theorem one is trying to prove here.

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December 1, 2016 at 11:14

Belated reply. I admit I don’t follow the proof (ie I follow all the steps except why it implies uncountability). But suppose you replace ℝ (set of real numbers in the proof) by ℂ (set of computable numbers, as defined by Turing and refined by Minsky and others) everywhere, and ask about ℂ\𝔸 (set of computable numbers that are not algebraic). Isn’t everything in the proof unchanged? But the computable numbers are countable.

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