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Simply definable well -orderings of the reals.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	Simply definable well -orderings of the reals./
Author:	Caicedo, Andres Eduardo.
Description:	1 online resource (102 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 65-08, Section: B.
Contained By:	Dissertations Abstracts International65-08B.
Subject:	Mathematics. -
Online resource:	click for full text (PQDT)
ISBN:	9780496527410

Simply definable well -orderings of the reals.
Caicedo, Andres Eduardo.

Simply definable well -orderings of the reals. - 1 online resource (102 pages)

Source: Dissertations Abstracts International, Volume: 65-08, Section: B.

Thesis (Ph.D.)--University of California, Berkeley, 2003.

Includes bibliographical references

In this thesis we explore the problem of obtaining simply definable well-orderings of the reals together with additional combinatorial structure on the continuum. Specifically: (1) We show that the following is consistent, without any restrictions in the large cardinal structure of the universe: "[special characters omitted] is real-valued measurable and there is a definable [special characters omitted]-well-ordering of the reals". More precisely: If there is a measurable, and [special characters omitted] holds below it, then there is a forcing extension satisfying the statement. (2) We present a similar argument, due to Woodin, that when applied to L[μ] produces a forcing extension where [special characters omitted] is real-valued measurable, and there is a [special characters omitted]-well-ordering of [special characters omitted]. The best result along these lines, due to Woodin, is that under appropriate large cardinal hypothesis, "[special characters omitted] is real-valued measurable and there is a definable [special characters omitted]-well-ordering of the reals" is Ω-consistent. (3) We introduce a strengthening of real-valued measurability, called real-valued hugeness, which implies the existence of many real-valued measurable cardinals and by results of Woodin, the determinacy of strong pointclasses. For example, [special characters omitted] holds. We show it is consistent that [special characters omitted] is real-valued huge, and present a result of Woodin proving that this property of [special characters omitted] contradicts the existence of any [special characters omitted]-well-ordering of [special characters omitted]. (4) We show that if there is no inner model with ω many strong cardinals, then there is a set forcing extension with a projective well-ordering of the reals (in fact, a [special characters omitted]-well-ordering, where n is the number of strong cardinals in K and, if n = 0 and V is not closed under sharps, then a [special characters omitted]-one). (5) We show that if there are no inner models with Woodin cardinals, and V is a finestructural model for a strong cardinal with a measurable above, then there is a forcing extension where [special characters omitted] +ψAC hold and where the reals admit a [special characters omitted]-well-ordering. We also start an investigation into the structure of inner models M of [special characters omitted] when a strong forcing axiom holds in V, and show: (1) Without loss of generality, [special characters omitted]. More precisely: If [special characters omitted] holds, then there is an inner model N, M ⊆ N, such that N |= [special characters omitted], and [special characters omitted] is inaccessible in N iff it is inaccessible in M. (2) Moreover, if [special characters omitted] holds and [special characters omitted] is a successor cardinal in M, then [special characters omitted] = ([special characters omitted]+)M, where cf([special characters omitted]) = ω, and [special characters omitted] fails in M. (3) In fact, whenever M is an inner model of [special characters omitted] correctly computing [special characters omitted] and such that [special characters omitted] = ([special characters omitted]+)M, where cfV([special characters omitted]) = ω, then (a) In M the approachability property fails at [special characters omitted] and there are no uniformly almost disjoint sequences for [special characters omitted], in particular cfM ([special characters omitted]) = ω. (b) V is not a weakly proper forcing extension of M, and there is no inner model of V that computes ω2 correctly where [special characters omitted] holds. In particular, if [special characters omitted] holds then there is a real r such that M [r] |= ¬[special characters omitted]. (c) Furthermore, if [special characters omitted], then [special characters omitted] fails in M, and [special characters omitted].

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2024

Mode of access: World Wide Web

ISBN: 9780496527410Subjects--Topical Terms:

527692
Mathematics.
Subjects--Index Terms:

CardinalsIndex Terms--Genre/Form:

554714
Electronic books.

Simply definable well -orderings of the reals.
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100 1 $a Caicedo, Andres Eduardo. $3 1472770
245 1 0 $a Simply definable well -orderings of the reals.
264 0 $c 2003
300 $a 1 online resource (102 pages)
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500 $a Source: Dissertations Abstracts International, Volume: 65-08, Section: B.
500 $a Publisher info.: Dissertation/Thesis.
500 $a Advisor: Steel, John;Woodin, W. Hugh.
502 $a Thesis (Ph.D.)--University of California, Berkeley, 2003.
504 $a Includes bibliographical references
520 $a In this thesis we explore the problem of obtaining simply definable well-orderings of the reals together with additional combinatorial structure on the continuum. Specifically: (1) We show that the following is consistent, without any restrictions in the large cardinal structure of the universe: "[special characters omitted] is real-valued measurable and there is a definable [special characters omitted]-well-ordering of the reals". More precisely: If there is a measurable, and [special characters omitted] holds below it, then there is a forcing extension satisfying the statement. (2) We present a similar argument, due to Woodin, that when applied to L[μ] produces a forcing extension where [special characters omitted] is real-valued measurable, and there is a [special characters omitted]-well-ordering of [special characters omitted]. The best result along these lines, due to Woodin, is that under appropriate large cardinal hypothesis, "[special characters omitted] is real-valued measurable and there is a definable [special characters omitted]-well-ordering of the reals" is Ω-consistent. (3) We introduce a strengthening of real-valued measurability, called real-valued hugeness, which implies the existence of many real-valued measurable cardinals and by results of Woodin, the determinacy of strong pointclasses. For example, [special characters omitted] holds. We show it is consistent that [special characters omitted] is real-valued huge, and present a result of Woodin proving that this property of [special characters omitted] contradicts the existence of any [special characters omitted]-well-ordering of [special characters omitted]. (4) We show that if there is no inner model with ω many strong cardinals, then there is a set forcing extension with a projective well-ordering of the reals (in fact, a [special characters omitted]-well-ordering, where n is the number of strong cardinals in K and, if n = 0 and V is not closed under sharps, then a [special characters omitted]-one). (5) We show that if there are no inner models with Woodin cardinals, and V is a finestructural model for a strong cardinal with a measurable above, then there is a forcing extension where [special characters omitted] +ψAC hold and where the reals admit a [special characters omitted]-well-ordering. We also start an investigation into the structure of inner models M of [special characters omitted] when a strong forcing axiom holds in V, and show: (1) Without loss of generality, [special characters omitted]. More precisely: If [special characters omitted] holds, then there is an inner model N, M ⊆ N, such that N |= [special characters omitted], and [special characters omitted] is inaccessible in N iff it is inaccessible in M. (2) Moreover, if [special characters omitted] holds and [special characters omitted] is a successor cardinal in M, then [special characters omitted] = ([special characters omitted]+)M, where cf([special characters omitted]) = ω, and [special characters omitted] fails in M. (3) In fact, whenever M is an inner model of [special characters omitted] correctly computing [special characters omitted] and such that [special characters omitted] = ([special characters omitted]+)M, where cfV([special characters omitted]) = ω, then (a) In M the approachability property fails at [special characters omitted] and there are no uniformly almost disjoint sequences for [special characters omitted], in particular cfM ([special characters omitted]) = ω. (b) V is not a weakly proper forcing extension of M, and there is no inner model of V that computes ω2 correctly where [special characters omitted] holds. In particular, if [special characters omitted] holds then there is a real r such that M [r] |= ¬[special characters omitted]. (c) Furthermore, if [special characters omitted], then [special characters omitted] fails in M, and [special characters omitted].
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2024
538 $a Mode of access: World Wide Web
650 4 $a Mathematics. $3 527692
653 $a Cardinals
653 $a Combinatorics
653 $a Well-orderings
655 7 $a Electronic books. $2 local $3 554714
690 $a 0405
710 2 $a ProQuest Information and Learning Co. $3 1178819
710 2 $a University of California, Berkeley. $3 1183587
773 0 $t Dissertations Abstracts International $g 65-08B.
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3105174 $z click for full text (PQDT)