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It is said that the intuitive meaning of $\mu$ is finite looping where as the intuitive meaning of $\nu$ is infinite looping in $\mu$ calculus. I understand this for finite systems, but why is this true in general?

Is there any theorem which proves this ?

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    $\begingroup$ Who says so? In particular, $\nu$ is new to me. $\endgroup$
    – Raphael ♦
    Nov 26, 2017 at 21:04
  • $\begingroup$ What context is this in? $\mu$ and $\nu$ are commonly used for least-fixed point and greatest-fixed point in PL-theory, is this what you mean? $\endgroup$
    – jmite
    Mar 27, 2018 at 1:03

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It is said that the intuitive meaning of μ is finite looping where as the intuitive meaning of ν is infinite looping in μ calculus

Here's a better phrasing: µ is about looping with a known bound on the number of iteration. For ν we do not necessarily know the bound, and there may in fact not even be one.

Is there any theorem which proves this ?

Theorems can not prove how apt an intuition is. The property you have in mind follows from the definitions and can be illustrated by some examples.

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  • $\begingroup$ How does one get this intuition in a system with uncountable number of states? How does this property follow from definitions? $\endgroup$
    – e_noether
    Nov 27, 2017 at 15:29
  • $\begingroup$ 1) No Turing-complete model has an uncountable number of states. 2) (Bounded) $\mu$ has known bounds, by definition. (Unbounded $\mu$ aka) $\nu$ has not. That is, if you're using the definitions I know. You don't say which you're using. $\endgroup$
    – Raphael ♦
    Nov 27, 2017 at 21:36

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