Definable Towers

2018/11/21 by V. Fischer, Fischer, V., J. Schilhan +1
Computer Science · Economics, Econometrics and Finance · Mathematics · #03E05 (Secondary) #03E15 #03E17 (Primary) #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Economic theories and models #FOS: Mathematics #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.1811.08775

openalex publication_date 2018/11/21 · openalex created_date 2022/09/02 · openalex updated_date 2026/07/28

Abstract

We study the definability of maximal towers and of inextendible linearly ordered towers (ilt's), a notion that is more general than that of a maximal tower. We show that there is, in the constructible universe, a Π¹₁ definable maximal tower that is indestructible by any proper Suslin poset. We prove that the existence of a Σ¹₂ ilt implies that ω₁ = ω₁^L. Moreover we show that analogous results hold for other combinatorial families of reals. We prove that there is no ilt in Solovay's model. And finally we show that the existence of a Σ¹₂ ilt is equivalent to that of a Π¹₁ maximal tower.

Related

• Hausdorff gaps and towers in P(ω)/Fin
• On Countable Stationary Towers
• Existence Theorem for Cumulative Universe Towers and Its Applications
• Metastability and the Furstenberg-Zimmer Tower II: Polynomial and Multidimensional Szemeredi's Theorem
• Towers in filters, cardinal invariants, and Luzin type families