Consider those phenomena which are influenced by their very announcement like prices, fashion. A very powerful method for "solving" these circularity is that of trying to view them as true proper equations in one indeterminate. Look at a singleton set. Here x is equal to itself, to the singleton set of itself. We can view this as an equation. With one indeterminate or more if for instance we take into account simultaneous systems of equations. And this is a way of writing the stream of Fibonacci numbers by just writing a set of two simultaneous equations with an operator which manipulates streams. But the fundamental question is the following: Do such equations have one unique solution? Are these virtuous circles? Or rather these equations do not have any solution at all. They are just vicious circles. Infinite regressive. And these expressions simply lead to paradoxes like liar's paradox. Here is a version of liar's paradox that perhaps people are not so familiar with. You see here there is not even self reference. The sentence P of n says something about all the sentences P of x for x greater than n. But nevertheless even if this is not a circular, a completely circular definition still it is a paradox. So as you see it appears that being able to define functions or defining concepts in terms of themselves, that is implicitly, makes life very easy. But it may happen that some of these circularities are indeed vicious, they do not define anything.

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