# How badly can the GCH fail globally?

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

In 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 $$kappa_n, n about it. They first force to get $$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.

## Related Questions

### Commutator estimates regarding pseudo-differential operators

0  Asked on December 15, 2020 by shaoyang-zhou

### English translation of Borel-Serre Le theoreme de Riemann-Roch?:

1  Asked on December 15, 2020

### Upper bound for an exponential sum involving characters of a finite field

1  Asked on December 14, 2020 by nahila

### Euler function summation

0  Asked on December 13, 2020 by andrej-leko

### A problem about an unramified prime in a Galois extension

1  Asked on December 9, 2020 by neothecomputer

### Reference request: discretisation of probability measures on $mathbb R^d$

1  Asked on December 9, 2020 by mb2009

### Smoothness of a variety implies homological smoothness of DbCoh

0  Asked on December 8, 2020 by dbcohsmoothness

### Reference for matrices with all eigenvalues 1 or -1

1  Asked on December 7, 2020

### Characterizations of groups whose general linear representations are all trivial

1  Asked on December 7, 2020 by qsh

### Continuous time Markov chains and invariance principle

0  Asked on December 6, 2020 by sharpe

### Continuity property for Čech cohomology

0  Asked on December 6, 2020 by xindaris

### Reference request: superconformal algebras and representations

0  Asked on December 6, 2020 by winawer

### Extending rational maps of nodal curves

1  Asked on December 5, 2020 by leo-herr

### How large is the smallest ordinal larger than any “minimal ordinal parameter” for any pair of an Ordinal Turing Machine and a real?

1  Asked on December 4, 2020 by lyrically-wicked

### Almost geodesic on non complete manifolds

1  Asked on December 4, 2020 by andrea-marino

### With Khinchine’s inequality, prove Fourier basis is unconditional in $L^{p}[0,1]$ only for $p=2$

0  Asked on December 3, 2020 by eric-yan

### Kähler manifolds deformation equivalent to projective manifolds

0  Asked on December 3, 2020 by user164740

### Duality of eta product identities: a new idea?

2  Asked on December 1, 2020 by wolfgang

### Is there a CAS that can solve a given system of equations in a finite group algebra $kG$?

2  Asked on December 1, 2020 by bernhard-boehmler

Get help from others!