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I was reading these notes and I recently asked:

Why does the fixed point theorem justify the existence of the factorial function?

that outlined the need of fixed points for justifying the definition of the (recursive) factorial function.

My question is now, how do we characterize all the fix points related to the operator defined based on the factorial function:

$$ \mathcal F (g)(n) = \left\{ \begin{array}{ll} 1 & \mbox{if } n = 0 \\ n * g(n-1) & \mbox{if } n > 0 \mbox{ and } g(n-1) \ne \bot\\ \bot & \mbox{if } n > 0 \mbox{ and } g(n-1) = \bot \end{array} \right. $$

where bottom $\bot$ means undefined (i.e. the elements that don't have a mapping to the codomain are mapped to this element, this happens because we have partial functions).

Usually fixed-points satisfy:

$$ \mathcal F (x) = x$$

if $F (x)$ was a polynomial then I'd know how to solve it, say: $f(x) = x^2 - 3x + 4$ then we solve $f(x) = x^2 - 3x + 4 = x$. Depending on the degree we might have many solutions and thus many fixed points.

However, when we are dealing with partial functions its really confusing. How do we characterize all the fixed points of the operator $\mathcal F$ defined based on the factorial function?

Also, once we have a fixed point that is not the least one, and we plug it in to the operator defined based on the factorial, how does that affect the definition of the factorial function? Is the factorial function still a well defined mathematical object?

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