An Order-Theoretic Characterization of Consistent Inductive Inference
Abstract
When can a learner make only finitely many prediction errors along every infinite sequence labeled by a fixed, unknown hypothesis? We characterize this form of consistency for arbitrary binary hypothesis classes in ZFC, without requiring a uniform mistake bound. The characterization uses a single linear order on finite realizable traces. Each trace selects its least subtrace, and the order must satisfy two conditions: conflicting traces select different subtraces, and the order is well-founded on the traces of each fixed target. These conditions induce a learner whose selected evidence decreases on every mistake. Conversely, a consistent learner yields such an order through canonical mistake transcripts and the Kleene--Brouwer ordering. The result provides a representation of consistent prediction by finite evidence, answering a question of Lu (2024).
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BibTeX
@misc{lu2026ordertheoretic,
title = {{An Order-Theoretic Characterization of Consistent Inductive Inference}},
author = {Zhou Lu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28551},
eprint = {2609.28551},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28551},
}
RIS
TY - UNPB
TI - An Order-Theoretic Characterization of Consistent Inductive Inference
AU - Zhou Lu
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28551
UR - https://arxiv.org/abs/2609.28551
AB - When can a learner make only finitely many prediction errors along every infinite sequence labeled by a fixed, unknown hypothesis? We characterize this form of consistency for arbitrary binary hypothesis classes in ZFC, without requiring a uniform mistake bound. The characterization uses a single linear order on finite realizable traces. Each trace selects its least subtrace, and the order must satisfy two conditions: conflicting traces select different subtraces, and the order is well-founded on the traces of each fixed target. These conditions induce a learner whose selected evidence decreases on every mistake. Conversely, a consistent learner yields such an order through canonical mistake transcripts and the Kleene--Brouwer ordering. The result provides a representation of consistent prediction by finite evidence, answering a question of Lu (2024).
ER -