Such as in the pair of statements РІР‚СљThe following sentence is false.РІР‚Сњ РІР‚СљThe preceding sentence is true.РІР‚Сњ The culprit in these paradoxes is self reference, or strange loopiness. So sets are either run of the mill, or self-swallowing, but we can define a set as The Set of All Run of the Mill Sets which must be neither self-swallowing or run of the mill. Or, they are self-swallowing, such as the set of all things that are no Joan of Arc. For example, the set of all walruses is not itself a walrus. RussellРІР‚в„ўs Paradox:Р’В Most sets are not members of themselves. The real conclusion is that the system is incomplete, there are true statements which its methods of proof are too weak to demonstrate.The Godel sentence G could be better written as РІР‚СљThis statement of number theory does not have any proof in the system of Principia Mathematica.РІР‚Сњ.РІР‚СљAll consistent axiomatic formulations of number theory include undecidable propositions.РІР‚Сњ GodelРІР‚в„ўs Incompleteness Theorem deals with strange loops, asserting that any attempt for a number theory to define itself will fail.Such as in the famous Escher paintings like the infinite staircase.Р”В®pimenides Paradox:Р’В Another form of strange loop, contained in statements like РІР‚СљThis statement is false.РІР‚Сњ It is a РІР‚Сљone step strange loop.РІР‚Сњ This is the first example in the book of a РІР‚СљStrange Loop,РІР‚СњР’В a loop whereby moving through it we unexpectedly find ourselves right back where we started. The transition is seamless, and allows it to continue rising forever. There are also more complicated canons, where the second voice may enter at a different pitch as well.Р”Р‡ugue:Р’В A fugue is like a canon, in that itРІР‚в„ўs based on one theme which gets played in different voices, but the notation is much less rigid and you can play with it more.Р”В«ach created an РІР‚Сљendlessly rising canon,РІР‚Сњ in which it ends and immediately restarts but a note up. In the basic canon, the first voice enters, and after a period, the second voice enters at the same beginning as the first, layering on top of each other.

The Offering has a three part fugue, a six part fugue, ten canons, and a trio sonata.Р”В¬anons:Р’В In a canon, a single theme is played against itself. In the Musical Offering, Bach includes an inscriptions whose first letters combine to spell РІР‚СљRICERCARРІР‚Сњ, and Italian word meaning РІР‚Сљto seekРІР‚Сњ

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