- #1

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Consider the following numbers:

0.1223334444...

0.112222333333...

0.111222222...

... (When numbers of two or more digits are encountered, they repeat in the same way, e.g. in the first number we would have ...99999999910101010101010101010.... )

These numbers are irrational, as they are non-repeating infinite decimals. They also form an countably infinite set, as they are infinite but can be listed.

However, for obvious reasons, ∏ is not included. Nor is e, √2, or many other irrationals.

Since the set of the irrationals contains more elements than a countable infinite set, it is uncountable. QED

Thanks for looking.