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Boolean-valued approach to the Lebesgue measure problem Sandberg Maitland, William
Abstract
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 .
Item Metadata
Title |
Boolean-valued approach to the Lebesgue measure problem
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
1977
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Description |
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 .
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Genre | |
Type | |
Language |
eng
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Date Available |
2010-02-21
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Provider |
Vancouver : University of British Columbia Library
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Rights |
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
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DOI |
10.14288/1.0080111
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Campus | |
Scholarly Level |
Graduate
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Aggregated Source Repository |
DSpace
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Rights
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.