Chapter 55: Possible worlds
551
Forcing with quotient algebras
Measurable spaces with negligibles; associated forcing notions;
representing names for members of {0,1}I; representing names
for Baire sets in {0,1}I; the usual measure on {0,1}I;
re-interpreting Baire sets in the forcing model; representing Baire
measurable functions; representing measure algebras; iterated
forcing; extending filters.
552
Random reals I
Random real forcing notions; calculating 2κ; b
and d; preservation of outer measure; Sierpinski sets;
cardinal functions of the usual measure on {0,1}λ;
Carlson's theorem on extending measures; iterated random real
forcing.
553
Random reals II
Rothberger's property; non-scattered compact sets; Haydon's
property; rapid p-point ultrafilters; products of ccc partially
ordered sets; Aronszajn trees; medial limits; universally
measurable sets.
554
Cohen reals
Calculating 2κ; Lusin sets; precaliber pairs of measure
algebras; Freese-Nation numbers; Borel liftings for Lebesgue
measure.
555
Solovay's construction of real-valued-measurable cardinals
Measurable cardinals are quasi-measurable after ccc forcing,
real-valued-measurable after random real forcing;
Maharam-type-homogeneity; covering number of
product measure; power set σ-quotient algebras can have
countable centering number or Maharam type; supercompact cardinals
and the normal measure axiom.
556
Forcing with Boolean subalgebras
Forcing names over a Boolean subalgebra; Boolean operations, ring
homomorphisms; when the subalgebra is regularly embedded; upper
bounds, suprema, saturation, Maharam type; quotient forcing;
Dedekind completeness; L0; probability algebras; relatively
independent subalgebras; strong law of large numbers; Dye's
theorem; Kawada's theorem; the Dedekind completion of the
asymptotic density algebra.
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