HEXAGONbeta

Ramsey’s Theorem for Pairs in Pincus’s Model XI

A mix of human-written and AI-generated textHuman understanding: some partsmath.LO — Logic

Contributed by Frank Gilson

Version 2 / Oct 22, 2026 / CC BY 4.0

Abstract

Howard and Rubin record Pincus’s ordered colored permutation model as N48, but do not determine whether Ramsey’s theorem for pairs holds there. We prove that N48 |= RT for pairs. The proof uses least finite supports and the homogeneity of Pincus’s ordered colored structure: after fixing all but one point of a support, the remaining point generates an infinite orbit whose unordered pairs form a single stabilizer orbit. Combining this with Pincus’s results on the dense linear ordering principle (DO), the Boolean Prime Ideal Theorem (BPI), and the failure of Part∞, together with Pincus’s transfer theorems, yields Con(ZF) ⇒ Con(ZF + RT_pairs + DO + BPI + ¬Part∞), where Part∞ states that every infinite set is the disjoint union of infinitely many infinite sets.

Provenance statement

I developed the research question, directed the investigation, selected and checked the relevant mathematical sources, and worked through the proof and transfer argument to the final form presented here. I reviewed the final mathematical argument and citations and understand the claims and reasoning in the paper. OpenAI’s ChatGPT was used extensively for mathematical exploration, proof development and auditing, literature searching, source verification, and drafting; Anthropic’s Claude was used for adversarial review of proofs, citations, and exposition. Meta AI and Apodex (Deep Solve mode) were also used for additional review during the earlier development of the work. All final mathematical and editorial decisions are mine.
AI provenance filesProvenance_Verification_Supplement_Pincus_Model_XI_v1.2_2026-10-21.md · 11 KB

Tools used

OpenAI
ChatGPT
Anthropic
Claude Opus
Meta
meta.ai
Apodex
apodex.ai Deep Solve

References

  1. A. Banerjee, A. Gopaulsingh, and Z. Molnár. On the Rainbow Ramsey theorem and the Canonical Ramsey Theorem for pairs without AC. arXiv:2401.02215v2 [math.LO], 2024.arXiv
  2. A. Blass. Ramsey's theorem in the hierarchy of choice principles. J. Symbolic Logic 42 (1977), no. 3, 387--390. DOI: 10.2307/2272866 (https://doi.org/10.2307/2272866).DOI
  3. J. Brot, M. Cao, and D. Fernández-Bretón. Finiteness classes arising from Ramsey-theoretic statements in set theory without choice. Ann. Pure Appl. Logic 172 (2021), no. 6, 102961. DOI: 10.1016/j.apal.2021.102961 (https://doi.org/10.1016/j.apal.2021.102961).DOI
  4. I. M. Dimitriou. Choiceless Grapher (Jeffrey) . Software repository, GitHub, ioannad/jeffrey , commit 75d800d (21 May 2025). Implication data supplied by P. Howard.
  5. P. Howard and J. E. Rubin. Consequences of the Axiom of Choice . Mathematical Surveys and Monographs, vol. 59, American Mathematical Society, Providence, RI, 1998. DOI: 10.1090/surv/059 (https://doi.org/10.1090/surv/059).DOI
  6. D. Pincus. Adding dependent choice. Ann. Math. Logic 11 (1977a), no. 1, 105--145. DOI: 10.1016/0003-4843(77)90011-0 (https://doi.org/10.1016/0003-4843(77)90011-0).DOI
  7. D. Pincus. Adding dependent choice to the prime ideal theorem. In R. O. Gandy and J. M. E. Hyland (eds.), Logic Colloquium '76 , Studies in Logic and the Foundations of Mathematics, vol. 87, North-Holland, Amsterdam, 1977b, pp. 547--565. DOI: 10.1016/S0049-237X(09)70445-8 (https://doi.org/10.1016/S0049-237X(09)70445-8).DOI
  8. D. Pincus. The dense linear ordering principle. J. Symbolic Logic 62 (1997), no. 2, 438--456. DOI: 10.2307/2275540 (https://doi.org/10.2307/2275540).DOI
  9. E. Tachtsis. On Ramsey's theorem and the existence of infinite chains or infinite anti-chains in infinite posets. J. Symbolic Logic 81 (2016), no. 1, 384--394. DOI: 10.1017/jsl.2015.47 (https://doi.org/10.1017/jsl.2015.47).DOI

Version history

  1. v1Submitted by Frank GilsonInitial depositSupersededOct 22, 2026
  2. v2Submitted by Frank GilsonSubmitted as a new versionCurrentOct 22, 2026