For what set theoretic reasons can a function not be included in its own domain?
Thanks
functionselementary-set-theory
asked 2012-12-01
user id:51341
21
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Can you please clarify what you mean by that? What exactly is a function to you? What does it mean for it to be included in its domain? – 2012-12-01
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What about the function $\emptyset:\emptyset\to\emptyset$? We do have $\emptyset\subset\emptyset$. Do you mean that a function $f:E\to F$ cannot be an element of $E$? – 2012-12-01
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Indeed, in Lambda Calculus, every element is a function with domain every element. (That doesn't help understand the set theoretic reasons for not allowing it, just pointing out that as a rule, it is possible to apply a function to itself...) – 2012-12-01
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@ThomasAndrews, you are identifying lambda expressions with functions, and the two are quite different things. One *can* attach semantics to lambda calculus where the meaning of lambda expressions is a function, but in any case the way to do this properly is somewhat complicated. It is probable not the best example to bring out in this context! :-) – 2012-12-01
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@MarianoSuárez-Alvarez well, it is the primary interpretation I know for Lambda calculus. Even the set theoretic definition of function is just a formality that we interpret as something like what we think of as a "function." As soon as I saw a question about "functions which are in their own domain," I thought of Lambda calculus. – 2012-12-01