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Can I say that humans in history named, recognized countable set of numbers (1,2..Pi,sqrt(2) ... etc)

the count of this set is named aleph-0 ?

2)

Do mathematicians investigate the "other" continuum set of numbers which is infinity bigger ?

Do we suspect, knows anything about that set like that .

For example I am naming pi_2019 number: pi-to-the power of pi ,to-the power of pi .... (times 2019 power it) (new number? I have named;)

3)

Did I move one number from one set to the other (named digits) ?