On page 230, 11 th line from the bottom, reference is made to "Soare", This should be changed to ... more On page 230, 11 th line from the bottom, reference is made to "Soare", This should be changed to "Richard Shore of M.I.f.".
This paper is a continuation of [3] and. presupposes the results proved there. The main theorem o... more This paper is a continuation of [3] and. presupposes the results proved there. The main theorem of this paper is Th,zorem 5.17, the co,,exing lemma for K, which asserts thttt, provided that there is no inner model with a measurable cardinal, then for every uncountable X tl~ere is Ye K with X= Y and X _c y. The reader will reeog~'~ise that this is an adap:afion of the covering lemma for L, and, indeed, that ff K should chance to be L (which happens precisely when 0 # does not exist) then tlqs statement reduces t(~ the covering lemma for L. It is in two senses the best possible result: firstly, if the~'e is an inner model with a measurable A. DoddRJense~ L[U, C] involves more work than we had realised; in the interests of [,re,5~zeing an accoant of the main result as speedily as possible ~ it is now t~ve yea~s since it was proved~we have decided to pre,~ent these last two results in a .3eparate paper "The covering lemma for L[U]". The authors would like to repeat t~eir thanks to all those whose help is acknowledged in the preface to [3], They are also indebted to New College, Oxford, who made possible the typing of the .aanu~cript. 1. Ptdhnhun'ies We shall not repeat the preliminary material from [3]. But a ~f~:w extra comments wil?, be in place. Two small errors, first. Claim 2 of Lemma 2.12 is false; this hall of the inequality is never used. And although the S,, hierarchy is alwa~ assumed transitive (e.g. in the proof of l.emma 5.19) this is not true for the us~ml finite basis of, say, [5]. It is not difficult to devise extra functions that will make it so. Corollary 3.21, though true, is insufficient for the results in this l:.aper. It is in fact the case, using the notation there, that ~(K)NX~(N~)= 9~(~:)n2~(/~,}) for all i, j, although this is not necessarily true if o~ is replaced by n; the procff follows from tKe techniques used to prove the result as stated.
COPYING AND REPRINTING. Individual readers of this publication, and nonprofit libraries acting fo... more COPYING AND REPRINTING. Individual readers of this publication, and nonprofit libraries acting for them are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews provided the customary acknowledgement of the sources is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Executive Director,
K.J. Devlin has extended Jensen's construction of a model of ZFC and CH without Souslin trees to ... more K.J. Devlin has extended Jensen's construction of a model of ZFC and CH without Souslin trees to a model without Kurepa trees either. We modify the construction again to obtain a model with these properties, but in addition, without Kurepa trees in ccc-generic extensions. We use a partially defined D-sequence, given by a fine structure lemma. We also show that the usual collapse of x Mahlo to ~02 will give a model without Kurepa trees not only in the model itself, but also in ccc-extensions.
We study type 1 premice equipped with supercomplete extenders. In this paper, we show that such p... more We study type 1 premice equipped with supercomplete extenders. In this paper, we show that such premice are normally iterable and all normal iteration trees of type 1 premice has a unique coÿnal branch. We give a construction of an K C type model using supercomplete type 1 extenders.
In this paper, we sketch the development of two important themes of modern set theory, both of wh... more In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory. §0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω + ω, … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture...
We show that either of the following hypotheses imply that there is an inner model with a proper ... more We show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal κ ≥ ℵ such that □κ and □(κ) fail. 2) There is a cardinal κ such that κ is weakly compact in the generic extension by Col(κ, κ+). Of special interest is 1) with κ = ℵ3 since it follows from PFA by theorems of Todorcevic and Velickovic. Our main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over Kc∥κ.
We show in ZFC that if there is no proper class inner model with a Woodin cardinal, then there is... more We show in ZFC that if there is no proper class inner model with a Woodin cardinal, then there is an absolutely definablecore modelthat is close toVin various ways.
On page 230, 11 th line from the bottom, reference is made to "Soare", This should be changed to ... more On page 230, 11 th line from the bottom, reference is made to "Soare", This should be changed to "Richard Shore of M.I.f.".
This paper is a continuation of [3] and. presupposes the results proved there. The main theorem o... more This paper is a continuation of [3] and. presupposes the results proved there. The main theorem of this paper is Th,zorem 5.17, the co,,exing lemma for K, which asserts thttt, provided that there is no inner model with a measurable cardinal, then for every uncountable X tl~ere is Ye K with X= Y and X _c y. The reader will reeog~'~ise that this is an adap:afion of the covering lemma for L, and, indeed, that ff K should chance to be L (which happens precisely when 0 # does not exist) then tlqs statement reduces t(~ the covering lemma for L. It is in two senses the best possible result: firstly, if the~'e is an inner model with a measurable A. DoddRJense~ L[U, C] involves more work than we had realised; in the interests of [,re,5~zeing an accoant of the main result as speedily as possible ~ it is now t~ve yea~s since it was proved~we have decided to pre,~ent these last two results in a .3eparate paper "The covering lemma for L[U]". The authors would like to repeat t~eir thanks to all those whose help is acknowledged in the preface to [3], They are also indebted to New College, Oxford, who made possible the typing of the .aanu~cript. 1. Ptdhnhun'ies We shall not repeat the preliminary material from [3]. But a ~f~:w extra comments wil?, be in place. Two small errors, first. Claim 2 of Lemma 2.12 is false; this hall of the inequality is never used. And although the S,, hierarchy is alwa~ assumed transitive (e.g. in the proof of l.emma 5.19) this is not true for the us~ml finite basis of, say, [5]. It is not difficult to devise extra functions that will make it so. Corollary 3.21, though true, is insufficient for the results in this l:.aper. It is in fact the case, using the notation there, that ~(K)NX~(N~)= 9~(~:)n2~(/~,}) for all i, j, although this is not necessarily true if o~ is replaced by n; the procff follows from tKe techniques used to prove the result as stated.
COPYING AND REPRINTING. Individual readers of this publication, and nonprofit libraries acting fo... more COPYING AND REPRINTING. Individual readers of this publication, and nonprofit libraries acting for them are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews provided the customary acknowledgement of the sources is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Executive Director,
K.J. Devlin has extended Jensen's construction of a model of ZFC and CH without Souslin trees to ... more K.J. Devlin has extended Jensen's construction of a model of ZFC and CH without Souslin trees to a model without Kurepa trees either. We modify the construction again to obtain a model with these properties, but in addition, without Kurepa trees in ccc-generic extensions. We use a partially defined D-sequence, given by a fine structure lemma. We also show that the usual collapse of x Mahlo to ~02 will give a model without Kurepa trees not only in the model itself, but also in ccc-extensions.
We study type 1 premice equipped with supercomplete extenders. In this paper, we show that such p... more We study type 1 premice equipped with supercomplete extenders. In this paper, we show that such premice are normally iterable and all normal iteration trees of type 1 premice has a unique coÿnal branch. We give a construction of an K C type model using supercomplete type 1 extenders.
In this paper, we sketch the development of two important themes of modern set theory, both of wh... more In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory. §0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω + ω, … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture...
We show that either of the following hypotheses imply that there is an inner model with a proper ... more We show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal κ ≥ ℵ such that □κ and □(κ) fail. 2) There is a cardinal κ such that κ is weakly compact in the generic extension by Col(κ, κ+). Of special interest is 1) with κ = ℵ3 since it follows from PFA by theorems of Todorcevic and Velickovic. Our main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over Kc∥κ.
We show in ZFC that if there is no proper class inner model with a Woodin cardinal, then there is... more We show in ZFC that if there is no proper class inner model with a Woodin cardinal, then there is an absolutely definablecore modelthat is close toVin various ways.
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