Is 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 t

Is 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?