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Completely Additive Liftings

Published online by Cambridge University Press:  15 January 2014

Ilijas Farah
Ilijas Farah*
Affiliation:
Department of Mathematics, York University, North York, on, CanadaM3J 1P3 Matematicki Institut, Kneza Mikaila 35 11000 Beograd, Yugoslavia, E-mail: ifarah@mathstat.yorku.ca
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Extract

The purpose of this communication is to survey a theory of liftings, as developed in author's thesis ([8]). The first result in this area was Shelah's construction of a model of set theory in which every automorphism of P(ℕ)/ Fin, where Fin is the ideal of finite sets, is trivial, or inother words, it is induced by a function mapping integers into integers ([33]). (It is a classical result of W. Rudin [31] that under the Continuum Hypothesis there are automorphisms other than trivial ones.) Soon afterwards, Velickovic ([47]), was able to extract from Shelah's argument the fact that every automorphism of P(ℕ)/ Fin with a Baire-measurable lifting has to be trivial. This, for instance, implies that in Solovay's model ([36]) all automorphisms are trivial. Later on, an axiomatic approach was adopted and Shelah's conclusion was drawn first from the Proper Forcing Axiom (PFA) ([34]) and then from the milder Open Coloring Axiom (OCA) and Martin's Axiom (MA) ([48], see §5 for definitions). Both shifts from the quotient P(ℕ)/ Fin to quotients over more general ideals P(ℕ)/I and from automorphisms to arbitrary ho-momorphisms were made by Just in a series of papers ([14]-[17]), motivated by some problems in algebra ([7, pp. 38–39], [43, I.12.11], [45, Q48]) and topology ([46, p. 537]).

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 4 , Issue 1 , March 1998 , pp. 37 - 54
Copyright
Copyright © Association for Symbolic Logic 1998

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