Sticks and clubs

1998/04/15 by Sakaé Fuchino, Saharon Shelah, Fuchino, Sakaé +3 · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.math/9804153

openalex publication_date 1998/04/15 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side-by-side product of partial orders which we call pseudo-product. Using such products, we give several generic extensions where some of these principles hold together with not CH and Martin's Axiom for countable p.o.-sets. An iterative version of the pseudo-product is used under an inaccessible cardinal to show the consistency of the club principle for every stationary subset of limits of omega₁ together with not CH and Martin's Axiom for countable p.o.-sets.

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• All spaces of countable spread can be small

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• On some variants of the club principle
• Pseudo-countable models
• Combinatorial principles from adding Cohen reals
• Combinatorial and hybrid principles for sigma-directed families of countable sets modulo finite
• Measuring club sequences, together with the Continuum Hypothesis