For what set theoretic reasons can a function not be included in its own domain?
Thanks
For what set theoretic reasons can a function not be included in its own domain?
Thanks
Corrected: A non-empty function can be a subset of its own domain. Let $x_0=0$, for each $n\in\omega$ let $x_{n+1}=\langle x_n,0\rangle$, and let $X=\{x_n:n\in\omega\}$; then
$f\triangleq\big\{\langle x_n,0\rangle:n\in\omega\big\}=\{x_{n+1}:n\in\omega\}\subseteq X=\operatorname{dom}f\;.$
One could use such a function to construct a set that violates the Axiom of Regularity.
Note: This answer assumes that "included" means "is an element of."