Exercises
[01R] (Solved on 2022-10-25) The axiom of good foundation (also called axiom of regularity) of the Zermelo–Fraenkel theory says that every non-empty set \(X\) contains an element \(y\) that is disjoint from \(X\); in formula
\[ ∀ X, X≠ ∅ ⇒ (∃ y\, (y∈ X)∧ (X∩ y=∅))~ ~ \](remember that every object in the theory is a set, so \(y\) is a set). Using this axiom prove these facts.
There is no set \(x\) that is an element of itself, that is, for which \(x∈ x\).
More generally there is no finite family \(x_ 1,\ldots x_ n\) such that \(x_ 1∈ x_ 2∈ \ldots ∈ x_ n ∈ x_ 1\).
There is also no \(x_ 1,\ldots x_ n,\ldots \) sequence of sets for which \(x_ 1∋ x_ 2 ∋ x_ 3 ∋ x_ 4\ldots \).
Solution 1