My aim is to discuss, on the basis of a historical survey, the question of the consistency of ZF set theory and of its large cardinal extensions. First, I explain the reasons why in the case of set theory neither model–theoretic nor proof–theoretic methods seem sufficient to come to grips with the problem of consistency. Then, I recall how set–theorists have dealt with the problem by means of large cardinals and inner models, with a reasonably confident attitude about consistency. Finally, I hint at finitary versions of Goedel sentence constructions (due to W. H. Woodin) which could allow a comparison between arithmetic and the extensions of set theory with respect to our knowledge of their consistency. I maintain that the main philosophical interest of the problem lies in the relationship between intuition and formalization.
The problem of consistency: ZF and large cardinals
BELLOTTI, LUCA
2005-01-01
Abstract
My aim is to discuss, on the basis of a historical survey, the question of the consistency of ZF set theory and of its large cardinal extensions. First, I explain the reasons why in the case of set theory neither model–theoretic nor proof–theoretic methods seem sufficient to come to grips with the problem of consistency. Then, I recall how set–theorists have dealt with the problem by means of large cardinals and inner models, with a reasonably confident attitude about consistency. Finally, I hint at finitary versions of Goedel sentence constructions (due to W. H. Woodin) which could allow a comparison between arithmetic and the extensions of set theory with respect to our knowledge of their consistency. I maintain that the main philosophical interest of the problem lies in the relationship between intuition and formalization.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
