TITLE: Can a countable set contain uncountably many infinite subsets such that the intersection of any two such distinct subsets is finite?
QUESTION [19 upvotes]: Can a countable set contain uncountably many infinite subsets such that the intersection of any two such distinct subsets is finite?

REPLY [4 votes]: For each $t\in(\frac1{10},1),$ take its decimal expansion (or one of its decimal expansions in case of non-uniqueness), and let $A_t$ be the set of all finite truncations of that decimal expansion, considered as positive integers. For example,
$$A_{\frac13}=\{3,\ 33,\ 333,\ 3333,\ 33333,\dots\},$$
$$A_{\pi-3}=\{1,\ 14,\ 141,\ 1415,\ 14159,\dots\}.$$