>>10922469# or |X| or card(X) would denote the cardinality of a set X.
Your set of all finite length sequences of 0's and 1's would be {0,1}^?0 , where {0,1}^?0 is the first countable ordinal and the operation is ordinal exponentiation.
But standard in more fields of math is to let Y^X denote the cardinal exponentiation, which classically is equal to the cardinality of the plain functions set X -> Y and by a minor abuse of notation, we might take Y^X to be that function space directly (in which case we don't write e.g. |Y^X| but let Y^X also denote the cardinality of that set)
We can write N here instead of ?0 or ?0 without confusion, but given that the question is so set theoretical, I'd assume people know what ?0 is and avoid the symbol associated with the arithmetic of the natural, which is of no use her at all.
Worth noting that the existence of the function space XY is constructively a weaker assumption than that of the cardinality and also worth pointing out that ordinal exponentiation is very different from cardinal exponentiation. E.g. your set {0,1}^?0 is countable, whereas {0,1}^?0, i.e. | ?0 -> {0,1} |, is not.
Also, in the conventional von Neumann model, limit cardinals such as ?0 are just their ordinal counterparts, e.g. ?0.
Anyway, I don't agree with
>>10919991, I'm not completely sure what the question is. I doubt OP cooked it up himself and the fact that we seemingly are to choose some obscure subset of the continuum X makes me think this is a difficult task.
