Artículos con la etiqueta ‘forcing’

Forcing axioms for λ -complete μ + -C.C

Por • 31 oct, 2013 • Category: Educacion

We note that some form of the condition “p_1, p_2 have a >_Q-lub in Q” is necessary in some forcing axiom for lambda-complete mu^+-c.c. forcing notions. We also show some versions are really stronger than others, a strong way to answer Alexie’s question of having P satisfying one condition but no P’ equivalent to P satisfying another. We have not looked systematically whether any such question (of interest) is open.



Understanding Preservation Theorems, Chapter VI of Proper and Improper Forcing, I

Por • 6 jun, 2013 • Category: Filosofía

This is an exposition of the first two sections of Chapter VI of Shelah’s book Proper and Improper Forcing. It covers various preservation theorems for CS iteration of proper forcing (omega-omega bounding, Sacks property, P-point property, etc.)



Category forcings, MM^{+++}, and generic absoluteness for the theory of strong forcing axioms

Por • 13 may, 2013 • Category: Crítica

We introduce a category whose objects are stationary set preserving complete boolean algebras and whose arrows are non-atomic complete homomorphisms with a stationary set preserving quotient. We show that the cut of this category at a rank initial segment of the universe of height a superhuge cardinal is a stationary set preserving partial order which forces MM^{++} and collapses its size to become the second uncountable cardinal. Next we argue that any of the known methods to produce a model of MM^{++} collapsing a superhuge cardinal to become the second uncountable cardinal produces a model in which the cutoff of this category at any rank initial segment of the universe of super huge height is forcing equivalent to a presaturated tower of normal filters.



Structural connections between a forcing class and its modal logic

Por • 5 ago, 2012 • Category: Crítica

The modal logic of forcing arises when one considers a model of set theory in the context of all its forcing extensions, interpreting necessity as “in all forcing extensions” and possibility as “in some forcing extension”. In this modal language one may easily express sweeping general forcing principles, such as the assertion that every possibly necessary statement is necessarily possible, which is valid for forcing, or the assertion that every possibly necessary statement is true, which is the maximality principle, a forcing axiom independent of but equiconsistent with ZFC (Stavi-V\”a\”an\”anen, Hamkins). Every definable forcing class similarly gives rise to the corresponding forcing modalities, for which one considers extensions only by forcing notions in that class. In previous work, we proved that if ZFC is consistent, then the ZFC-provably valid principles of the class of all forcing are precisely the assertions of the modal theory S4.2.



Forcing consequences of PFA together with the continuum large

Por • 12 mar, 2012 • Category: Crítica

We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using our method we prove that the forcing axiom for the class of all the small finitely proper posets is compatible with a large continuum.



EL PROBLEMA DEL CONTINUO DESPUÉS DE COHEN (1964-2004)

Por • 19 jun, 2011 • Category: Crítica

En este trabajo se expone en que consiste el nuevo axioma llamado Martin Máximo Acotado (BMM)1, el cual es un axioma que puede considerarse “natural” en cierto sentido y que junto con la teoría ZFE decide el problema del continuo de Cantor. El llamado Axioma de Martin (AM) es un conocido enunciado relacionado con la topología, la combinatoria infinita y el forcing, planteado por Donald Martin en 1970. En 1988 Foreman, Magidor y Shelah, formularon una versión fuerte maximal de AM y lo llamaron Martin Máximo (MM). También demostraron la consistencia de MM relativa a la existencia de un cardinal supercompacto. BMM es una modificación acotada de MM que resulta más débil y que decide el problema del continuo, en el sentido de que el cardinal del continuo es aleph 2