Large Cardinals and The Axiom of Determinacy
The consistency of the axiom of determinacy is closely related to the question of the consistency of large cardinal axioms. By a theorem of Woodin, the consistency of Zermelo–Fraenkel set theory without choice (ZF) together with the axiom of determinacy is equivalent to the consistency of Zermelo–Fraenkel set theory with choice (ZFC) together with the existence of infinitely many Woodin cardinals. Since Woodin cardinals are strongly inaccessible, if AD is consistent, then so are an infinity of inaccessible cardinals.
Moreover, if to the hypothesis of an infinite set of Woodin cardinals is added the existence of a measurable cardinal larger than all of them, a very strong theory of Lebesgue measurable sets of reals emerges, as it is then provable that the axiom of determinacy is true in L(R), and therefore that every set of real numbers in L(R) is determined.
Read more about this topic: Axiom Of Determinacy
Famous quotes containing the words large and/or axiom:
“In looking back over the college careers of those who for various reasons have been prominent in undergraduate life ... one cannot help noticing that these men have nearly always shown from the start an interest in the lives of their fellow students. A large acquaintance means that many persons are dependent on a man and conversely that he himself is dependent on many. Success necessarily means larger responsibilities, and responsibilities mean many friends.”
—Franklin D. Roosevelt (1882–1945)
“The writer who neglects punctuation, or mispunctuates, is liable to be misunderstood.... For the want of merely a comma, it often occurs that an axiom appears a paradox, or that a sarcasm is converted into a sermonoid.”
—Edgar Allan Poe (1809–1845)