The well-known Liar's Paradox "This statement is false" leads to a recursive contradiction: If the statement is interpreted to be true then it is actually false, and if it is interpreted to be false then it is actually true. The statement is a paradox where neither truth value can be assigned to it.
However, "This statement is true" also leads to a paradox where either truth value can be assigned to it with equal validity. If the statement is perceived to be true then it is actually true, and if the statement is perceived to be false then it is actually false.
These two statements demonstrate two different classes of paradox.
The same paradox states exist in set theory. Consider "The set of all sets that do not contain themselves" leads to the former paradox (neither solution is valid), and "the set of all sets that do contain themselves" leads to the latter paradox (either solution is valid.)
My question is: How many classifications of paradox exist? Is there any development in classifying types of paradoxes and applying them to mathematical logic, computer science, and set theory? What implications would classes of paradoxes have on Gödel's incompleteness theorems--could a system that allows and classifies paradoxes be demonstrably consistent?