0
$\begingroup$

I want to make such a construction:

Let $X$ be an infinite set. Put:

$Y_0 = X \\ y_0 \in Y_0$

$Y_1 = Y_0 - \{y_0\} \\ y_1 \in Y_1$

...

$Y_n = Y_{n-1} - \{y_{n-1}\} \\ y_n \in Y_n$

...

for every natural $n$.

How to prove the existence of function $y_n$? I could use AC, but for that I need family of sets $Y_n$, and from the other hand, to construct that family of sets I need to have function $y_n$.

The chicken and the egg problem? How to solve that?

3 Answers 3