Other Problems
The notation can be complicated, especially as in the previous example, and abbreviations are often employed when context indicates the nature of a variable. For example:
- {x : x > 0}, in a context where the variable x is used only for real numbers, indicates the set of all positive real numbers;
- {p/q : q ≠ 0}, in a context where the variables p and q are used only for integers, indicates the set of all rational numbers; and
- {S : S does not belong to S}, in a context where the variable S is used only for sets, indicates the set of all sets that don't belong to themselves.
As the last example shows, such an abbreviated notation again might not denote an actual nonparadoxical set, unless there is in fact a set of all objects that might be described by the variable in question.
Read more about this topic: Set-builder Notation
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