This is a non-precisely formulated question recently come to mind:
How to investigate the cardinality of the set of all subsequences of an arbitrarily given sequence? Or can we possibly determine the cardinality of the set of all subsequences of any arbitrary sequence? (the sequence may be in an (at most) countable set or in an uncountable set)
This question is non-precisely formulated because that there may be at least two ways to define "equivalent" subsequences: (1) two subsequences $(x_{n(j)})$ and $(x_{m(j)})$ are equivalent iff $x_{n(j)}=x_{m(j)}$ for all $j\in\mathbb{N}$ (that is, they have the same value/term in corresponding places of two subsequences), or (2) two subsequences $(x_{n(j)})$ and $(x_{m(j)})$ are equivalent iff $n(j)=m(j)$ for all $j\in\mathbb{N}$ (that is, the two subsequences are chosen from values/terms of the same index in the original sequence.)
In the above two definitions of "equivalent" subsequences, can we and how to investigate the cardinality of the set of all subsequences? Is someone interested to help formulating this question precisely (and maybe edit this post) and to share your viewpoints about this question?