## Notre Dame Journal of Formal Logic

### Closed Maximality Principles and Generalized Baire Spaces

Philipp Lücke

#### Abstract

Given an uncountable regular cardinal $\kappa$, we study the structural properties of the class of all sets of functions from $\kappa$ to $\kappa$ that are definable over the structure $\langle\mathrm{{H}}(\kappa^{+}),\in\rangle$ by a $\Sigma_{1}$-formula with parameters. It is well known that many important statements about these classes are not decided by the axioms of $\mathrm{{ZFC}}$ together with large cardinal axioms. In this paper, we present other canonical extensions of $\mathrm{{ZFC}}$ that provide a strong structure theory for these classes. These axioms are variations of the Maximality Principle introduced by Stavi and Väänänen and later rediscovered by Hamkins.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 60, Number 2 (2019), 253-282.

Dates
Accepted: 17 February 2017
First available in Project Euclid: 8 May 2019

https://projecteuclid.org/euclid.ndjfl/1557281186

Digital Object Identifier
doi:10.1215/00294527-2019-0004

Mathematical Reviews number (MathSciNet)
MR3952233

#### Citation

Lücke, Philipp. Closed Maximality Principles and Generalized Baire Spaces. Notre Dame J. Formal Logic 60 (2019), no. 2, 253--282. doi:10.1215/00294527-2019-0004. https://projecteuclid.org/euclid.ndjfl/1557281186

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