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A method of constructing iterated forcing notions that has a scope of applications similar to Shelah's oracle-c.c. is presented. This method yields a consistency result on homomorphisms of quotient algebras of the Boolean algebra 9'(w). Also, it is shown to be relatively consistent with ZFC that the Boolean algebra of Lebesgue measurable subsets of the unit interval has no projective lifting.

A modification of Shelah’s oracle-c.c.\ with applications