The number of r-element subsets of a set of cardinality n is given by $_nC_r$, with is equivalent to the factorial expression $$\frac{n!}{r!(n-r)!}$$ Well, whenever I see a factorial I get the urge to analytically continue it with the gamma function, which would make the expression above $$\frac{Γ(n+1)}{Γ(r+1)Γ(n-r+1)}$$ Which gave me the idea to find the number of $i$-element subsets in a $1/2$-element set, or $$\frac{Γ(3/2)}{Γ(i+1)Γ(3/2-i)}$$ Which, according to Wolfram Alpha, has a value of approximately $2.62 + 1.28i$. What I'm wondering is how a set of non-integer cardinality could be interpreted? Has anybody done any work on this subject? Or have I stumbled upon a new and probably useless area of set theory?
2026-03-26 19:01:58.1774551718
Analytic continuation of the cardinality of a set: -1/2 element sets? Sets with an imaginary number of elements?
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