Definition
A strange loop is a hierarchical structure in which moving through what feels like one direction returns to the starting point — but at a different conceptual level. Coined by Douglas Hofstadter in Gödel, Escher, Bach (1979), the term names the structural pattern that links Bach's Endlessly Rising Canon (which modulates upward through six keys and ends one octave higher), Escher's Drawing Hands (two hands drawing each other into existence), and Gödel's self-referential sentence (which encodes a claim about its own provability).
A strange loop appears whenever a system's rules are rich enough to let a lower level refer to, and partially constitute, a higher level — so that traversing the hierarchy in one direction reliably brings you back to where you started while crossing a level boundary.
The simplest specimen is the Liar paradox: "This sentence is false." Evaluate it as true and it becomes false; evaluate it as false and it becomes true. Chasing the truth value upward through the levels of the assertion sends you straight back to the bottom. The Liar is not a quirk of grammar but a structural feature of any system expressive enough to talk about itself — which is exactly why Bertrand Russell's theory of types tried, unsuccessfully, to banish self-reference from the foundations of mathematics.
Why it matters
How it works
A strange loop requires three ingredients. First, a hierarchy with distinct levels — pitch / key in music, figure / frame in art, formula / proof in mathematics, neuron / symbol in cognition. Second, a set of rules at the lower level that are expressive enough to describe or affect the higher level — Bach's notes can modulate keys; Gödel-numbers encode formulas about formulas; neurons firing can update the self-symbol that determines which neurons fire. Third, an act of self-application: the description at the lower level is applied to itself, producing a fixed point where the level boundary becomes a closed loop.
In Hofstadter's later book I Am a Strange Loop (2007) he makes the case that the human self is exactly this structure realized in brain tissue: a self-symbol embedded in the symbol-level activity that also drives the brain's behavior. Consciousness is, on this view, what such a strange loop feels like from inside.
Turing's halting problem is the computational analogue of the same pattern: no general algorithm can decide, for every program-input pair, whether the program halts. The proof constructs a self-referential program that halts if and only if it does not halt — the identical level-crossing loop that Gödel built in arithmetic, instantiated in computation instead.