3 ms·
Sets are defined by their members. If you have a nonempty set, you must have defined a member when you proved it was nonempty. Take any convenient one. This wo
by hgsgm 4y ago
Sets are defined by their members. If you have a nonempty set, you must have defined a member when you proved it was nonempty. Take any convenient one.
This works for any finite or countable family, because you can repeat the process for each set, any set you can think of will be processed eventually.
But for an uncountable set, almost every set will never be processed, and almost every set is impossible to describe in finite time (per set!), unless you just assume that you can do it on one magical oracle bulk selection.
- rtpg 4y agoIs the correct way to see this is that logic has to be specified in a finite number of terms? For countable families there’s some sort of meta logic like “for the Nth element I induct to that”? I believe to understand that idea (since it comes up in real analysis to some extent, basically acknowledging there isn’t really infinite sums except as limits of finite sums), but is there a specific sort of text that lays out this idea specifically?