Are there any subsets of $\ \mathbb{R}\ $ that have been investigated, but where it was not determined whether the set is countable or uncountable?

Asked Jun 25 '21 at 12:04

Active Jun 25 '21 at 12:04

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Are there any subsets of $\ \mathbb{R}\ $ that have been investigated, but where it was not determined whether the set is countable or uncountable? In particular, I am thinking of a subset of $\ \mathbb{R}\ $ where some condition was imposed on the decimal expansions of the numbers in the set.

I have seen this thread, but I am not very knowledgeable on cardinals/ordinals. I was looking for a condition on the decimal expansion of the real numbers.

Or is it the case that such a set would translate to an open problem in mathematics / set theory?

asked Jun 25 '21 at 12:04

Adam Rubinson

20,052

What is the meaning of where it was not determined, are you asking for a set whose Countabilty is an open problem ? – Vivaan Daga Jun 25 '21 at 12:11
Essentially yes. But the problem must have arisen from investigating a condition on the decimal expansions of real numbers. – Adam Rubinson Jun 25 '21 at 12:35

Are there any subsets of $\ \mathbb{R}\ $ that have been investigated, but where it was not determined whether the set is countable or uncountable?

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