## Journal of Symbolic Logic

- J. Symbolic Logic
- Volume 68, Issue 2 (2003), 527- 550.

### A simple maximality principle

#### Abstract

In this paper, following an idea of Christophe Chalons,
I propose a new kind of forcing axiom, the *Maximality
Principle*, which asserts that any sentence φ holding in
some forcing extension $V^{\P}$ and all subsequent extensions
V^{\P*\Qdot} holds already in V. It follows, in fact, that
such sentences must also hold in all forcing extensions of V.
In modal terms, therefore, the Maximality Principle is expressed
by the scheme
$(\possible\necessaryφ)\implies\necessaryφ$, and is
equivalent to the modal theory S5. In this article, I prove
that the Maximality Principle is relatively consistent with \ZFC.
A boldface version of the Maximality Principle, obtained by
allowing real parameters to appear in φ, is
equiconsistent with the scheme asserting that $V_{δ}\elesub V$
for an inaccessible cardinal δ, which in turn is
equiconsistent with the scheme asserting that $\ORD$ is Mahlo.
The strongest principle along these lines is
$\necessary\MPtilde$, which asserts that $\MPtilde$ holds in V
and all forcing extensions. From this, it follows that 0^{#}
exists, that x^{#} exists for every set x, that projective
truth is invariant by forcing, that Woodin cardinals are
consistent and much more. Many open questions remain.

#### Article information

**Source**

J. Symbolic Logic, Volume 68, Issue 2 (2003), 527- 550.

**Dates**

First available in Project Euclid: 11 May 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.jsl/1052669062

**Digital Object Identifier**

doi:10.2178/jsl/1052669062

**Mathematical Reviews number (MathSciNet)**

MR1976589

**Zentralblatt MATH identifier**

1056.03028

#### Citation

Hamkins, Joel David. A simple maximality principle. J. Symbolic Logic 68 (2003), no. 2, 527-- 550. doi:10.2178/jsl/1052669062. https://projecteuclid.org/euclid.jsl/1052669062