archery

The whole point of this sentence is to make clear what the whole point of this sentence is.

Just for a beginning, before the knots begin to appear. Let’s consider: What is a knot? If it is in the plane, then it is not. If it is not then it is a knot. Let G be defined by the equation Gx = F(xx). Then GG=F(GG).
Thus, if J=GG then J=F(J) for any F! (This is the fixed point theorem of Church and Curry in the untyped lambda calculus). The fixed point theorem gets you very quickly to paradox. For example, let AB mean “A is a member of B”. Let Rx = not(xx). (That is, R is a member of x only if x is not a member of x.) Then RR = not(RR). R is a member of R only if it is not a member of R! R’s Self-Membership is in a state of doubt. Now imagine a simple loop of rope. Allow that when a bit of line passes underneath another bit of line, we shall say that the underpassing bit “belongs” to the overpassing bit. Membership by underpassage. The simple loop is then an empty “knot set”. Put a twist in the loop and it underpasses itself. The singly twisted loop is a member of itself. Loop and twisted loop are topologically equivalent. Hence, speaking {topo}logically, the simple loop is both a member of itself and not a member of itself. By this simple twist of logic, the paradox becomes a phenomenon of three dimensional space.

What we mean by a set is a collection of things satisfying some criterion. In this regard, consider the set of all animals. Obviously this set is not an animal and therefore is not contained within itself. On the other hand, consider the complementary set consisting of all things that are not animals. In this case this set does contain itself since a set is not an animal. Define another set, say R, such that R contains all sets that do not contain themselves. Then the question for you is: does the set R contain itself?
If it does, then by definition of the set R, R does not contain itself. If R does not contain itself, then R does contain itself by construction. Either way, we are left with a contradiction. This marks the departure from naive set theory to ZFC.