2019年5月20日 · (I've actually seen it claimed, by a competent non-set-theorist, that V=L prevents Banach-Tarski - by virtue of implying that every set of reals is Borel.) And there are even deeper issues. For one thing, it's not even clear that V=L is actually first-order expressible!

2017年5月15日 · W. Hugh Woodin. The Axiom V = Ultimate-L. In preparation, 2016. W. Hugh Woodin. Fine Structure at the ...

2023年5月24日 · In his lecture Prof. Hugh Woodin have stated that "Assuming the omega conjecture, then from the universal laws of V_omega + 2 one can compute universal laws for V_omega + 3, ..., V_delta where delta is some large cardinal ..., The entire hierarchy of laws collapse". He stated that this cannot possibly be the correct conception of truth.

The main reason for this is that one cannot directly use forcing to prove that a statement is independent of ZFC+V=L, because no nontrivial forcing extension can satisfy this theory. Simply put, forcing extensions never satisfy V=L. Thus, our main tool for proving independence over ZFC does not work at all for proving independence over ZFC+V=L.

2019年1月8日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

$\begingroup$ For your question about satisfaction, you don't need to translate it separately into an arithmetic statement---the general phenomenon is that when a real codes a countable structure, then quantifying over that structure amounts to quantifying over …

2014年12月13日 · Using the ``instrumentalist dodge'' of Steel/Hamkins, it seems that we can "cheat" to obtain stronger large cardinal axioms within the confines of $\operatorname{ZF}+(V = L)$, such as "there is a transitive model of the theory ZFC+"$0^\sharp$ exists.""

2023年1月8日 · This is not a full answer, but I found it interesting to notice that if we relax the c.e. requirement somewhat, then there is a sweeping positive answer.

2023年6月29日 · For example, under $\text{AD}^+ + V=L(P(ℝ))$, the Ultrapower Axiom holds in HOD (Generalizations of the Ultrapower Axiom by Goldberg, 2023). Large cardinal axioms in the long extender region give order to many aspects of the cumulative hierarchy.

2023年5月28日 · Good day to you all, I would like to ask a question about relation between Prof. H. Woodin V = Ultimate L and a concept of set theoretical multiverse as proposed by Prof. Hamkins. If V = Ultimate L turns out to be true, does that mean that the whole set theoretic multiverse collapses?