explain the fixed point idea

The Yanofsky paper explains is in quite gentle a manner.

The key insight behind Lawvere's abstract framework to paradoxa is the the general existence of certain fixpoints.

Definition. We say that a set B has the fixpoint property if any function f:B→B has a fixpoint (i.e. f b = b for some b in B).

With this convenient definition, we are now ready to state and prove Lawvere's ridiculously simple and yet great theorem.

Theorem (Lawvere). If e:A→(A→B) is a surjective function, then B has the fixed point property.

The proof is quite easy. Let e:A→(A→B) be surjective. We have to show that B has the fixpoint property. That means for every f:B→B there is b∈B such that f(b)=b. Choose a function f:B→B and define the function g:A→B by setting

   a ↦ f (e a a)

As e is surjective, there must be a0∈A such that

   e a0 = g.

But then immediately

   f (g a0) = f (e a0 a0) = g a0

Hence g a0 is a fixpoint of f's.

Now many/most paradoxa are special cases of Lawvere's theorem. But for each paradox, the specific functions involved are a bit different. Let's look at an example.

Theorem (Cantor). There is no surjection e:A→Pow(A).

To see why this is true, note that Pow(A) is isomorphic to A→Bool. But there is a function on Bool that has no fixpoints, for example negation ¬:Bool→Bool, contradicting Lawvere's theorem.

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What is the intuition behind Lawvere's theorem? At first worrying about A→(A→Bool) is a bit surprising. What does that have to do with paradoxa and self-reference?

Well, what does it mean that A can speak about itself? To approach an answer, we could maybe first ask a simpler question: what does it mean to speak about A? How about this for an answer: to speak about A means to say something about A's elements. What does it mean to say something about A's elements? Maybe stating whether any given element a∈A has a property of interest? But what's a property? Easy: a property of A's elements is a function

   p:A→Bool

But we don't want just a fixed property, we want arbitrary properties. To do so, we have to consider the function space

   A→Bool

And how can we turn this into self-reference? What if each element a in A corresponded to a property over A? In other words, (with a lot of handwaving) self-reference means the existence of a surjective function

   A→(A→Bool)

The next step is to wonder: why Bool? Why not any old set? Note that Cantor's theorem continues to hold if we replace Bool with a larger set, but does not hold, if B in A→(A→B) has cardinality 1. What Bool and larger sets have in common is that we can rearrange them, i.e. there is a permuation that doesn't have a fixpoint.