Here is my final attempt at rewording these things, saying essentially the same thing you're saying (and I'm sure you're sick of reading my deleted comments by now):
1. Any element in T that is also a member of some set in class S is the same thing as the union of all sets arrived at by taking the intersection of T and that set (when that set is in class S).
2. Every element of T and any elements that are in every set in class S is the same thing as the intersection of all sets arrived at by taking the union of T and that set (when that set is in class S).
In set theory, there's no such thing as a 'plain old set'. Every set is either empty or else it contains other sets. "So then how on earth do you get numbers?" I hear you think...
Well, you say that 0 is the empty set. Then having defined the numbers from 0 up to n, you define n+1 = {0,...,n}.
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1. Any element in T that is also a member of some set in class S is the same thing as the union of all sets arrived at by taking the intersection of T and that set (when that set is in class S).
2. Every element of T and any elements that are in every set in class S is the same thing as the intersection of all sets arrived at by taking the union of T and that set (when that set is in class S).
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In set theory, there's no such thing as a 'plain old set'. Every set is either empty or else it contains other sets. "So then how on earth do you get numbers?" I hear you think...
Well, you say that 0 is the empty set. Then having defined the numbers from 0 up to n, you define n+1 = {0,...,n}.
It gets more complicated after that...
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