## Notre Dame Journal of Formal Logic

### Adding a Nonreflecting Weakly Compact Set

Brent Cody

#### Abstract

For $n\lt \omega$, we say that the$\Pi ^{1}_{n}$-reflection principle holds at $\kappa$ and write $\operatorname{Refl}_{n}(\kappa )$ if and only if $\kappa$ is a $\Pi ^{1}_{n}$-indescribable cardinal and every $\Pi ^{1}_{n}$-indescribable subset of $\kappa$ has a $\Pi ^{1}_{n}$-indescribable proper initial segment. The $\Pi ^{1}_{n}$-reflection principle $\operatorname{Refl}_{n}(\kappa )$ generalizes a certain stationary reflection principle and implies that $\kappa$ is $\Pi ^{1}_{n}$-indescribable of order $\omega$. We define a forcing which shows that the converse of this implication can be false in the case $n=1$; that is, we show that $\kappa$ being $\Pi ^{1}_{1}$-indescribable of order $\omega$ need not imply $\operatorname{Refl}_{1}(\kappa )$. Moreover, we prove that if $\kappa$ is $(\alpha +1)$-weakly compact where $\alpha \lt \kappa ^{+}$, then there is a forcing extension in which there is a weakly compact set $W\subseteq\kappa$ having no weakly compact proper initial segment, the class of weakly compact cardinals is preserved and $\kappa$ remains $(\alpha +1)$-weakly compact. We also formulate several open problems and highlight places in which standard arguments seem to break down.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 60, Number 3 (2019), 503-521.

Dates
Accepted: 7 November 2017
First available in Project Euclid: 11 June 2019

https://projecteuclid.org/euclid.ndjfl/1560218426

Digital Object Identifier
doi:10.1215/00294527-2019-0014

Mathematical Reviews number (MathSciNet)
MR3985624

Subjects
Primary: 03E35: Consistency and independence results
Secondary: 03E05: Other combinatorial set theory 03E55: Large cardinals

#### Citation

Cody, Brent. Adding a Nonreflecting Weakly Compact Set. Notre Dame J. Formal Logic 60 (2019), no. 3, 503--521. doi:10.1215/00294527-2019-0014. https://projecteuclid.org/euclid.ndjfl/1560218426

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