The electric buzzer and the governor of a steam engine (both examples taken from Gregory Bateson’s book “Mind and Nature”) are cycles, but cycles operating in time. In static (timeless) logic, they produce either nonsense or paradox. They are related to the Cretan liar paradox in its several versions, always put in the language of logic. In mathematics (a logical deductive system), this paradox is an important part of Gödel’s proof of incompleteness. It seems that the human mind can discover non-theorem type of truths, unlike formal algebra and unlike computer algorithms.

Is this because time plays no part in mathematics, which is seen as an “eternal” deductive system? But one might think that time does play a part in computer operations; there are temporal sequences in algorithms, and even a clock in the computer hardware. But that is an error: computer algorithms, like the steps in an algebraic proof, should not be seen as temporal sequences. They are only the steps in syllogisms. And computer hardware is composed only of “logic gates”. (The “clock” is only a mechanical convenience.)

A computer asked to “solve” the Cretan liar paradox will give alternating “true” and “false” answers, ad infinitum. This is a bifurcation of truth; yet only the first bifurcation (an oscillation), not chaos, which involves an infinite number of bifurcations (but following each other at rapidly converging intervals, so that chaos can be reached in finite time). It is a case of limited symmetry-breaking. Fuzzy logic solves the paradox by calling it half-true and half-false. As usual, we generalize by permitting fractions, i.e. creating new “numbers” as we need them.

The buzzer, the steam-engine governor, and the Cretan liar paradox are examples of real-world cycles between two possibilities. Penrose’s three worlds (Platonic, physical, mental) is an example of a three-pronged intransitive cycle, which is also a paradox. But intransitive paradoxes abound, like the game of “paper-stone-scissors” or the “voter’s paradox”. They only seem to go against logic, because logic is too narrow. They are perfectly explainable, in a different way in each case. The “Great Wrap-Around” of God and the Universe is the supreme example. The Snake is able to eat its tail.