TY - THES
AU - Sandberg Maitland, William
PY - 1977
TI - Boolean-valued approach to the Lebesgue measure problem
KW - Thesis/Dissertation
LA - eng
M3 - Text
AB - We let:
ZF = the Zermelo-Fraenkel axioms of set theory without the Axiom of Choice„(AC) . ZFC = ZF + AC .
I = " There exists an inaccessible cardinal " .
ψ = " Every set of reals definable from a count able sequence of ordinals is Lebesgue measurable ".
DC = the Axiom of Dependent Choices.
LM = " Every set of reals is Lebesgue measurable
In 1970, Solovay published a proof by forcing of the following relative consistency result:
Theorem
If there exists a model M of ZFC + I, then
there exist extensions M [G] and N of M such that:
(a) M [G] |= ZFC + ψ.
(b) N I= ZF + DC + LM .
Boolean-valued techniques are used here to retrace Solovay's proof on a different foundation and prove the following result:
Theorem
Let IK be a non-minimal standard transitive
model of ZFC + I. Then:
(a) IK |= there is a model of ZFC + ψ
(b) IK |= there is a model of ZF + DC + LM .
N2 - We let:
ZF = the Zermelo-Fraenkel axioms of set theory without the Axiom of Choice„(AC) . ZFC = ZF + AC .
I = " There exists an inaccessible cardinal " .
ψ = " Every set of reals definable from a count able sequence of ordinals is Lebesgue measurable ".
DC = the Axiom of Dependent Choices.
LM = " Every set of reals is Lebesgue measurable
In 1970, Solovay published a proof by forcing of the following relative consistency result:
Theorem
If there exists a model M of ZFC + I, then
there exist extensions M [G] and N of M such that:
(a) M [G] |= ZFC + ψ.
(b) N I= ZF + DC + LM .
Boolean-valued techniques are used here to retrace Solovay's proof on a different foundation and prove the following result:
Theorem
Let IK be a non-minimal standard transitive
model of ZFC + I. Then:
(a) IK |= there is a model of ZFC + ψ
(b) IK |= there is a model of ZF + DC + LM .
UR - https://open.library.ubc.ca/collections/831/items/1.0080111
ER - End of Reference