Descartes employed radical doubt until he reached the point where he could doubt the fact that he was doubting. This was the paradox that stopped him in his tracks.
This is from The Dream of Reality: Heinz von Foerster’s Constructivism by Lynn Segal (1986).
A paradox is a statement that is false when it is true and true when it is false. Paradox can occur whenever statements are self-referential. For instance: 1) This statement is false. 2) I am lying. 3) Please ignore this notice. 4) It is forbidden to forbid.
A paradoxical assertion is indeterminable or undecidable, which means a theorem or proposition cannot be proved or disproved. The paradox “it is forbidden to forbid” cannot be established as true or false.
Each statement comments on itself. . . . Why do logicians object to paradox? . . . Logicians work with declarative statements called propositions. . . . Aristotle taught that if a proposition makes sense it must be either true or false. . . Paradox renders a proposition’s truth value indeterminable. Paradoxical statements or propositions are neither true nor false.Segal
Autopoietic systems are self-referential systems. There is a re-entry of a distinction into a distinction that allows the system to observe itself.
The elements of the system can only connect with elements of the same system (which also have to be the same kind of elements); the system produces its own elements. It produces itself; it is not produced by anything outside of itself. It is a causa sui. This means that when a system observes itself it has no beginning and no end. As far the system is concerned, it has always existed and will always exist. It cannot conceive of its own ending because each operation presupposes a subsequent operation. Nor can it conceive of its own beginning because every operation presupposes a previous operation. Thus a human being cannot imagine her/his own death. There always has to be something coming next. Beginning and end can only be observed from outside the system.
Similarly, if we want to talk about language, then we need a meta-language. This is how we dissolve a paradox. But this is really a fiction because a metalanguage is still just language and it is still just talking about itself. In order for the discussion of language to carry on, we need to ignore the fact that we’re using language to talk about language. That is to say, we have to ignore the distinction (language/metalanguage) that we are using. If, in this case, we can’t put our use of language in a blind spot, we will tie ourselves up in knots. When we use language to observe (discuss, think about) language, we move to second-order observation.
Russell’s paradox has to do with logical classes and the logical elements the classes may contain. A class is a logical collection of objects that share a defined property. If one defines a class of books–all books past, present, and future–one can logically separate all objects in the universe into two classes: those that have class membership and those that don’t. If we allow a self-referential statement by asking if the class is a book, no paradox occurs. The class of books is not a book. (Segal )
So, the class of books does not contain itself. But what if we progress to a set of all sets that do not contain themselves as an element? The only way to dissolve this paradox is to move to a higher level.