Strong Measure Zero Sets on 2^κ for κ Inaccessible

2019/08/28 by Nick Steven Chapman, Chapman, Nick Steven, Johannes Philipp Schürz +1
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1908.10718

openalex publication_date 2019/08/28 · openalex created_date 2023/04/12 · openalex updated_date 2026/07/28

Abstract

We investigate the notion of strong measure zero sets in the context of the higher Cantor space 2^κ for κ at least inaccessible. Using an iteration of perfect tree forcings, we give two proofs of the relative consistency of |2^κ| = κ⁺⁺ + ∀ X ⊆ 2^κ: X is strong measure zero if and only if |X| ≤ κ⁺. Furthermore, we also investigate the stronger notion of stationary strong measure zero and show that the equivalence of the two notions is undecidable in ZFC.

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