The discussion centers on the proof of the Knaster-Tarski Theorem, specifically addressing the properties of a monotone function \( F: P(A) \to P(A) \) and the invariant sets under \( F \). The user questions the correctness of their proof, particularly the assertion that \( C = \bigcup_{X \subseteq A} F(F(X)) = F(X) \). The main point of contention is the implication that if \( B \in C \), then \( B \in X \) for some subset \( X \subseteq A \) such that \( X \subseteq F(X) \).
PREREQUISITESMathematicians, students of advanced mathematics, and researchers interested in fixed-point theorems and their applications in theoretical computer science and logic.
Evgeny.Makarov said:What is your question?
So, if I understand correctly, $$C=\bigcup_{X\subseteq A}F(F(X))=F(X)$$. But then it is not clear why $B\in C$ implies $B\in X$ for some $X\subseteq A$ such that $X\subseteq F(X)$.Fermat said:$C$ be the union of sets whose image is invariant under F