It’s not possible to implement the Banach Tarski paradox unless you subscribe to the Axiom of choice, this is clearly the easiest way to get Sarah to stop trying to bring back that dead hamster. You carefully think through a sensitive way of bringing this up because you know that she’s a huge fan of this axiom.
“So, if I remember right, for this to work it needs to be so that for any set of X of nonempty sets, there exists a choice function f that is defined on X and maps each set of X to an element of that set”
“Yes the axiom of choice, ”
She did her homework. You quickly dive into a Wikipedia hole to catch up.
“But correct me if I’m wrong but…” you stumble for words “Wouldn’t the sets of Dini X and the mapped set of X not be disjoint, uhhh, I just don’t think it fits Zermelo-Fáenkel set theory,” you try and quote from your smart phone.
“Well look at this,” she points to an equation on her white board. “Dini’s family of sets is SOOOO capable of this choice
∀x (∃o (o ∈ x ∧ ¬∃n (n ∈ o)) ∨ ∃a ∃b ∃c (a ∈ x ∧ b ∈ x ∧ c ∈ a ∧ c ∈ b ∧ ¬(a = b)) ∨ ∃c ∀e (e ∈ x → ∃a (a ∈ e ∧ a ∈ c ∧ ∀b ((b ∈ e ∧ b ∈ c) → a = b))))
You mutter the words “That doesn’t guarantee a partition of a set X the existence of a subset C,” but you know you don’t know what you’re talking about.
She had you there. You soon realized you didn’t know enough about the Axiom of choice to refute anything she was talking about and was much more read on the subject. You know in your heart that she’s wrong but you don’t know enough about the axiom to prove her one way or another and you can tell with her, she’s not going to be convinced whatever you come up with.
What do you do?
Dini cannot be Paradoxically decomposed when he is already normally decomposing.
The Banach Tarski paradox only works on spheres, the transforms don’t work on hamster shapes