Tarski's Fixed Point Theorem

Tarski’s Fixed Point Theorem states that, in a well-structured set with an order (like numbers or sets) and an operation that consistently moves things upward or downward, there will always be a point where applying the operation doesn’t change the point itself—a fixed point. Specifically, if the operation is monotone (preserves order), then least and greatest fixed points exist. This theorem is fundamental in areas like logic, computer science, and mathematics because it guarantees that certain processes or functions will reach a stable state or solution when iterated, which is key in reasoning about programs, semantics, and recursive definitions.