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The Axiom of Choice does not allow to "choose" a choice function, it only says that a choice function exists. To show that a choice function for a single nonempty set exists, you do not need to "choose" an element in the set, it is enough to show that at least one element exists (i.e. the set is nonempty).

What did you mean by "choosing" an element from a set? ;)

The Axiom of Choice does not allow to "choose" a choice function, it only says that a choice function exists. To show that a choice function for a single nonempty set exists, you do not need to "choose" an element in the set, it is enough to show that at least one element exists (i.e. the set is nonempty). Every element in the set will give a different choice function.

What did you mean by "choosing" an element from a set? ;)

The Axiom of Choice does not allow to "choose" a choice function, it only says that a choice function exists. To show that a "choice function" for a single nonempty set exists, you do not need to "choose" an element in the set, it is enough to show that at least one element exists (i.e. the set is nonempty).

What did you mean by "choosing" an element from a set? ;)

The Axiom of Choice does not allow to "choose" a choice function, it only says that a choice function exists. To show that a choice function for a single nonempty set exists, you do not need to "choose" an element in the set, it is enough to show that at least one element exists (i.e. the set is nonempty).

What did you mean by "choosing" an element from a set? ;)

The Axiom of Choice does not allow to "choose" a choice function, it only says that a choice function exists. To show that a "choice function" for a single nonempty set exists, you do not need to "choose" an element in the set, it is enough to show that at least one element exists.

What did you mean by "choosing" an element from a set? ;)

The Axiom of Choice does not allow to "choose" a choice function, it only says that a choice function exists. To show that a "choice function" for a single nonempty set exists, you do not need to "choose" an element in the set, it is enough to show that at least one element exists (i.e. the set is nonempty).

What did you mean by "choosing" an element from a set? ;)