Artículos con la etiqueta ‘Paul J. Cohen’

On ground model definability

Por • 6 dic, 2013 • Category: Opinion

Laver, and Woodin independently, showed that models of ZFC are uniformly definable in their set-forcing extensions, using a ground model parameter. We investigate ground model definability for models of fragments of ZFC , particularly of ZF+DC δ and of ZFC − , and we obtain both positive and negative results. Generalizing the results of Laver and Woodin, we show that models of ZF+DC δ are uniformly definable in their set-forcing extensions by posets admitting a gap at δ , using a ground model parameter. In particular, this means that models of ZF+DC δ are uniformly definable in their forcing extensions by posets of size less than δ . We also show that it is consistent for ground model definability to fail for models of ZFC − of the form H κ + . Using forcing, we produce a ZFC universe in which there is a cardinal κ>>ω such that H κ + is not definable in its Cohen forcing extension. As a corollary, we show that there is always a countable transitive model of ZFC − violating ground model definability. These results turn out to have a bearing on ground model definability for models of ZFC . It follows from our proof methods that the hereditary size of the parameter that Woodin used to define a ZFC model in its set-forcing extension is best possible.



La prueba de Cohen: la independencia de la hipótesis del continuo respecto a los otros axiomas de la teoría de conjuntos

Por • 2 dic, 2013 • Category: Filosofía

A finales de 1963 y principios de 1964 se produjo un acontecimiento de importancia fundamental en la historia de la matemática y la lógica, con la publicación del trabajo de Paul J. Cohen, The independence of the Continuum Hypothesis, o sobre la independencia de la hipótesis del continuo respecto a los otros axiomas de la teoría de conjuntos. En dicho trabajo Cohen realizó la hazaña extraordinaria de dividir la teoría de los conjuntos, precisamente de la misma manera como Lobachevski consiguió establecer en 1826 la división de la geometría en euclidiana y no-euclidiana.