Consistent and inconsistent generalizations of Martin's Axiom, weak square, and weak Chang's Conjecture

David Asperó; Nutt Tananimit

doi:10.1142/S0219061324500211

Consistent and inconsistent generalizations of Martin's Axiom, weak square, and weak Chang's Conjecture

David Asperó
, Nutt Tananimit

Algebra, Number Theory, Logic, and Representations (ANTLR)

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

34 Downloads (Pure)

Abstract

We prove that the forcing axiom MA _ℵ2 ^1.5(stratified) implies □ _ω1 _ω1. Using this implication, we show that the forcing axiom MM _ℵ2(ℵ ₂-c.c.) is inconsistent. We also derive weak Chang’s Conjecture from MA _ℵ21.5(stratified) and use this second implication to give another proof of the inconsistency of MM _ℵ2(ℵ ₂-c.c.).

Original language	English
Article number	2450021
Number of pages	31
Journal	Journal of Mathematical Logic (jml)
Volume	25
Issue number	03
Early online date	Jul 2024
DOIs	https://doi.org/10.1142/S0219061324500211
Publication status	Published - Dec 2025

Keywords

Weak square
forcing axiom failures
generalizations of Martin’s Axiom
weak Chang’s Conjecture
ℵ1.5-c.c. with respect to families of models with additional properties
ℵ -c.c. with respect to families of models with additional properties

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abstract = "We prove that the forcing axiom MA ℵ2 1.5(stratified) implies □ ω1 ω1. Using this implication, we show that the forcing axiom MM ℵ2(ℵ 2-c.c.) is inconsistent. We also derive weak Chang{\textquoteright}s Conjecture from MA ℵ21.5(stratified) and use this second implication to give another proof of the inconsistency of MM ℵ2(ℵ 2-c.c.).",

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T1 - Consistent and inconsistent generalizations of Martin's Axiom, weak square, and weak Chang's Conjecture

AU - Asperó, David

AU - Tananimit, Nutt

PY - 2025/12

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N2 - We prove that the forcing axiom MA ℵ2 1.5(stratified) implies □ ω1 ω1. Using this implication, we show that the forcing axiom MM ℵ2(ℵ 2-c.c.) is inconsistent. We also derive weak Chang’s Conjecture from MA ℵ21.5(stratified) and use this second implication to give another proof of the inconsistency of MM ℵ2(ℵ 2-c.c.).

AB - We prove that the forcing axiom MA ℵ2 1.5(stratified) implies □ ω1 ω1. Using this implication, we show that the forcing axiom MM ℵ2(ℵ 2-c.c.) is inconsistent. We also derive weak Chang’s Conjecture from MA ℵ21.5(stratified) and use this second implication to give another proof of the inconsistency of MM ℵ2(ℵ 2-c.c.).

KW - Weak square

KW - forcing axiom failures

KW - generalizations of Martin’s Axiom

KW - weak Chang’s Conjecture

KW - ℵ1.5-c.c. with respect to families of models with additional properties

KW - ℵ -c.c. with respect to families of models with additional properties

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DO - 10.1142/S0219061324500211

M3 - Article

SN - 0219-0613

VL - 25

JO - Journal of Mathematical Logic (jml)

JF - Journal of Mathematical Logic (jml)

IS - 03

M1 - 2450021

ER -

Consistent and inconsistent generalizations of Martin's Axiom, weak square, and weak Chang's Conjecture

Abstract

Keywords

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