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On a Theorem of Zarach Concerning the Global Pattern of Cardinals in Generic Extensions of Models of ZFC
Author(s) -
Pelletier Donald H.
Publication year - 1977
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/9.2.168
Subject(s) - mathematics , argument (complex analysis) , function (biology) , discrete mathematics , combinatorics , pure mathematics , biochemistry , chemistry , evolutionary biology , biology
Let M ⊨ ZFC and let F :o n M→o n Mbe a function on the ordinals of M . For which such F will there exist a model N of ZFC with the same ordinals as M such that N ⊨ γ is a cardinal if and only if M ⊨ (∃α) (γ = ℵ F (α) )? A partial answer to this question was given in a theorem of Zarach. We present here a counter‐example and correction to this theorem. Our argument invokes results of Drake concerning the collapse of weak cardinal powers in certain generic extensions.