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

Logic

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

Abstract

We prove that the forcing axiom $\MA^{1.5}_{\aleph_2}(\mbox{stratified})$ implies $\Box_{\omega_1, \omega_1}$.
Using this implication, we show that the forcing axiom $\MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$ is inconsistent. We also derive weak Chang's Conjecture from $\MA^{1.5}_{\aleph_2}(\mbox{stratified})$ and use this second implication to give another proof of the inconsistency of $\MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$.

Original language	English
Article number	2450021
Number of pages	31
Journal	Journal of Mathematical Logic (jml)
Early online date	Jul 2024
DOIs	https://doi.org/10.1142/S0219061324500211
Publication status	E-pub ahead of print - Jul 2024

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

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10.1142/S0219061324500211

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

abstract = "We prove that the forcing axiom $\MA^{1.5}_{\aleph_2}(\mbox{stratified})$ implies $\Box_{\omega_1, \omega_1}$. Using this implication, we show that the forcing axiom $\MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$ is inconsistent. We also derive weak Chang's Conjecture from $\MA^{1.5}_{\aleph_2}(\mbox{stratified})$ and use this second implication to give another proof of the inconsistency of $\MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$. ",

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author = "David Asper{\'o} and Nutt Tananimit",

year = "2024",

month = jul,

doi = "10.1142/S0219061324500211",

language = "English",

journal = "Journal of Mathematical Logic",

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TY - JOUR

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

AU - Asperó, David

AU - Tananimit, Nutt

PY - 2024/7

Y1 - 2024/7

N2 - We prove that the forcing axiom $\MA^{1.5}_{\aleph_2}(\mbox{stratified})$ implies $\Box_{\omega_1, \omega_1}$. Using this implication, we show that the forcing axiom $\MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$ is inconsistent. We also derive weak Chang's Conjecture from $\MA^{1.5}_{\aleph_2}(\mbox{stratified})$ and use this second implication to give another proof of the inconsistency of $\MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$.

AB - We prove that the forcing axiom $\MA^{1.5}_{\aleph_2}(\mbox{stratified})$ implies $\Box_{\omega_1, \omega_1}$. Using this implication, we show that the forcing axiom $\MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$ is inconsistent. We also derive weak Chang's Conjecture from $\MA^{1.5}_{\aleph_2}(\mbox{stratified})$ and use this second implication to give another proof of the inconsistency of $\MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$.

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

UR - http://www.scopus.com/inward/record.url?scp=85197638498&partnerID=8YFLogxK

U2 - 10.1142/S0219061324500211

DO - 10.1142/S0219061324500211

M3 - Article

JO - Journal of Mathematical Logic

JF - Journal of Mathematical Logic

SN - 0219-0613

M1 - 2450021

ER -

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

Abstract

Keywords

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