idea

Closed systems: what a loop costs

If the top level loops back to the bottom, the system effectively vouches for its own consistency, and the second theorem says a consistent system cannot do that. So a closed loop has to give up one of three things.

Consistency: Priest and the dialetheists accept some true contradictions. Total truth: Kripke’s theory lets a system talk about its own truth, but sentences like the liar stay neither true nor false. The ladder itself: Aczel’s non-well-founded sets contain themselves.

Hofstadter’s tangled hierarchy is the intuitive version: you climb levels and end up where you started. The Gödel sentence is already a tiny closed loop, one step long.

Connects to

• three views
• which cost?
• Hofstadter

Sources

1. Kripke, S. (1975). Outline of a Theory of Truth. Journal of Philosophy 72(19), 690–716.