It’s known that we have global failures of GCH—for example, where $forall lambda(2^lambda = lambda^{++})$—given suitable large cardinal axioms.

My question is whether we can have global failures of GCH where there is a weakly inaccessible cardinal between $lambda$ and $2^lambda$ for each $lambda$. Similarly, whether we can have a cardinal fixed point between $lambda$ and $2^lambda$. I’d also be interested in whether $2^lambda$ can be weakly inaccessible/a cardinal fixed point, for every $lambda$.

MathOverflow Asked by Sam Roberts on January 24, 2021

1 AnswersIn the Foreman-Woodin model The generalized continuum hypothesis can fail everywhere. for each infinite cardinal $kappa, 2^kappa$ is weakly inaccessible.

This answers your last question. The answer to the first two questions can be yes as well. In the case of Foreman-Woodin model, they start with a supercompact $kappa=kappa_0$ and infinitely many inaccessibles $kappa_n, n<omega,$ about it. They first force to get $2^{kappa_n}=kappa_{n+1}$ preserving $kappa$ supercompact, and this is reflected below for all cardinals. So if for example each $kappa_n$ is measurable, then what you get in the final model is that for each infinite cardinal $lambda, 2^lambda$ has been measurable in $V$, in particular there are both weakly inaccessible and cardinal fixed points between $lambda$ and $2^lambda.$

See also the paper A model in which every Boolean algebra has many subalgebras by Cummings and Shelah, where they build a model in which for each infinite cardinal $kappa, 2^kappa$ is weakly inaccessible and $Pr(2^kappa)$ holds. Here $Pr(lambda)$ is in some sense a large cardinal property (for example it holds if $lambda$ is a Ramsey cardinal). For its definition see the paper.

Correct answer by Mohammad Golshani on January 24, 2021

0 Asked on December 15, 2020 by shaoyang-zhou

1 Asked on December 15, 2020

1 Asked on December 14, 2020 by nahila

2 Asked on December 10, 2020 by yada

1 Asked on December 9, 2020 by neothecomputer

ac commutative algebra algebraic number theory nt number theory ra rings and algebras

1 Asked on December 9, 2020 by mb2009

0 Asked on December 8, 2020 by dbcohsmoothness

ag algebraic geometry derived algebraic geometry derived categories homological algebra

1 Asked on December 7, 2020

1 Asked on December 7, 2020 by qsh

algebraic groups geometric group theory gr group theory linear algebra rt representation theory

0 Asked on December 6, 2020 by sharpe

0 Asked on December 6, 2020 by xindaris

at algebraic topology cohomology limits and colimits sheaf cohomology

0 Asked on December 6, 2020 by winawer

lie superalgebras mp mathematical physics reference request rt representation theory vertex algebras

1 Asked on December 5, 2020 by leo-herr

ag algebraic geometry algebraic curves birational geometry blow ups

1 Asked on December 4, 2020 by lyrically-wicked

1 Asked on December 4, 2020 by andrea-marino

calculus of variations connections differential topology geodesics transversality

0 Asked on December 3, 2020 by eric-yan

0 Asked on December 3, 2020 by user164740

ag algebraic geometry at algebraic topology complex geometry dg differential geometry gt geometric topology

2 Asked on December 1, 2020 by wolfgang

big picture co combinatorics modular forms special functions

2 Asked on December 1, 2020 by bernhard-boehmler

computer algebra finite groups gr group theory ra rings and algebras rt representation theory

Get help from others!

Recent Questions

- MouseLook Script “Pops” back to the last value when the script is enabled after being disabled or destroyed
- Unity app crashes when using unmodified custom Android manifest (didn’t find class “UnityPlayerActivity”)
- How do i draw a ray in unity
- How to test consistency of responses?
- How can I understand these variograms?

Recent Answers

- eric_kernfeld on How to test consistency of responses?
- Justin Markwell on Unity app crashes when using unmodified custom Android manifest (didn’t find class “UnityPlayerActivity”)
- kjetil b halvorsen on How to test consistency of responses?
- Philipp on How do i draw a ray in unity
- DMGregory on MouseLook Script “Pops” back to the last value when the script is enabled after being disabled or destroyed

© 2022 AnswerBun.com. All rights reserved.