Stronger Forms of Determinacy
Several set-theoretic principles about determinacy stronger than Borel determinacy are studied in descriptive set theory. They are closely related to large cardinal axioms.
The axiom of projective determinacy states that all projective subsets of a Polish space are determined. It is known to be unprovable in ZFC but relatively consistent with it and implied by certain large cardinal axioms. The existence of a measurable cardinal is enough to imply over ZFC that all analytic subsets of Polish spaces are determined.
The axiom of determinacy states that all subsets of all Polish spaces are determined. It is inconsistent with ZFC but equiconsistent with certain large cardinal axioms.
Read more about this topic: Borel Determinacy Theorem
Famous quotes containing the words stronger and/or forms:
“When a strong man, fully armed, guards his castle, his property is safe. But when one stronger than he attacks him and overpowers him, he takes away his armor in which he trusted and divides his plunder.”
—Bible: New Testament, Luke 11:21.22.
“I would urge that the yeast of education is the idea of excellence, and the idea of excellence comprises as many forms as there are individuals, each of whom develops his own image of excellence. The school must have as one of its principal functions the nurturing of images of excellence.”
—Jerome S. Bruner (20th century)