# An Order-Theoretic Characterization of Consistent Inductive Inference

arXiv: https://arxiv.org/abs/2609.28551 · HTML: https://mlsanity.com/paper/2609.28551/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 9 references that are in the map.
- First posted on arXiv 23 Sep 2026.

## Builds on (most cited first, 3 of 3)

- [A Theory of Universal Learning](https://mlsanity.com/paper/2011.04483/) (2020)
- [Stable Sample Compression Schemes: New Applications and an Optimal SVM Margin Bound](https://mlsanity.com/paper/2011.04586/) (2020)
- [Computable universal online learning](https://mlsanity.com/paper/2510.18352/) (2025)

## Abstract (arXiv)

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).