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Reflection Properties of Ordinals in Generic Extensions

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Juan P. Aguilera, Corey Bacal Switzer

Logic · Submitted 21 Nov 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove, amongst other things, that adding any number of Cohen or random reals, or forcing with Sacks forcing or any lightface Borel weakly homogeneous ccc forcing notion cannot change such reflection properties.

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arXiv ID2311.12533 (version 1)
Pith ID7HFPN5X7, Pith's permanent record of this paper and its review
CitationCorey Bacal Switzer, Juan P. Aguilera. "Reflection Properties of Ordinals in Generic Extensions." arXiv:2311.12533 (2023). https://pith.science/paper/7HFPN5X7
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,089 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.39 · human participants 0.01 · controlled comparison 0.17 · randomized assignment 0.01 · states quantitative result 0.39 · compares with prior results 0.55 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We study the question of when a given countable ordinal $\alpha$ is $\Sigma^1_n$- or $\Pi^1_n$-reflecting in models which are neither $\mathsf{PD}$ models nor the constructible universe, focusing on generic extensions of $L$. We prove, amongst other things, that adding any number of Cohen or random reals, or forcing with Sacks forcing or any lightface Borel weakly homogeneous ccc forcing notion cannot change such reflection properties. Moreover we show that collapse forcing increases the value of the least reflecting ordinals but, curiously, to ordinals which are still smaller than the $\omega_1$ of $L$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study the question of when a given countable ordinal $\alpha$ is $\Sigma^1_n$- or $\Pi^1_n$-reflecting in models which are neither $\mathsf{PD}$ models nor the constructible universe, focusing on generic extensions of $L$. contribution 0.68 0.77 0.07 asserted 0.50
s1 We prove, amongst other things, that adding any number of Cohen or random reals, or forcing with Sacks forcing or any lightface Borel weakly homogeneous ccc forcing notion cannot change such reflection properties. result 1.00 0.94 0.48 asserted 1.00
s2 Moreover we show that collapse forcing increases the value of the least reflecting ordinals but, curiously, to ordinals which are still smaller than the $\omega_1$ of $L$. result 1.00 0.96 0.20 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We study the question of when a given countable ordinal $\alpha$ is $\Sigma^1_n$- or $\Pi^1_n$-reflecting in models which are neither $\mathsf{PD}$ models nor the constructible universe, focusing on generic extensions of $L$. We prove, amongst other things, that adding any number of Cohen or random reals, or forcing with Sacks forcing or any lightface Borel weakly homogeneous ccc forcing notion cannot change such reflection properties. Moreover we show that collapse forcing increases the value of the least reflecting ordinals but, curiously, to ordinals which are still smaller than the $\omega_1$ of $L$.

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