Douglas Hofstadter

Mature human brains are constantly trying to reduce the complexity of what they perceive, and this means that they are constantly trying to get unfamiliar, complex patterns made of many symbols that have been freshly activated in concert to trigger just one familiar pre-existing symbol (or a very small set of them). In fact, that’s the main business of human brains — to take a complex situation and to put one’s finger on what matters in it, to distill from an initial welter of sensations and ideas what a situation really is all about. To spot the gist. To Spot, the gist, however, doesn’t much matter, and the gist certainly doesn’t matter one whit to the flea on Spot’s wagging tail.

Implicit in the concept of Strange Loops is the concept of infinity, since what else is a loop but a way of representing an endless process in a finite way?

For instance, one level in a drawing might clearly be recognizable as representing fantasy or imagination; another level would be recognizable as reality. These two levels might be the only explicitly portrayed levels. But the mere presence of these two levels invites the viewer to look upon himself as part of yet another level; and by taking that step, the viewer cannot help getting caught up in Escher's implied chain of levels, in which, for any one level, there is always another level above it of greater "reality", and likewise, there is always a level below, "more imaginary" than it is.

All consistent axiomatic formulations of number theory
include undecidable propositions.
This is the pearl.

Godel had the insight that a statement of number theory could be about a statement of number theory (possibly even itself), if only numbers could somehow stand for statements.

Principia Mathematica was a mammoth exercise in exorcising Strange Loops from logic, set theory, and number theory. The idea of their system was basically this. A set of the lowest "type" could contain only "objects" as membersnot sets. A set of the next type up could only contain objects, or sets of the lowest type. In general, a set of a given type could only contain sets of lower type, or objects. Every set would belong to a specific type. Clearly, no set could contain itself because it would have to belong to a type higher than its own type.

It is very important when studying formal systems to distinguish working within the system from making statements or observations about the system.

Without doubt, Strange Loops involving rules that change themselves, directly or indirectly, are at the core of intelligence.