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 2^{κ}; **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 2^{κ}; 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; *L*^{0}; 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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Revised 13.11.10