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Large cardinals and definable counterexamples to the continuum hypothesis

Matthew Foreman, Menachem Magidor
<span title="">1995</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bnojym2hjzcgnpa4wixi2axhnq" style="color: black;">Annals of Pure and Applied Logic</a> </i> &nbsp;
In this paper we consider whether L([w) has "enough information" to contain a counterexample to the continuum hypothesis.  ...  We believe this question provides deep insight into the difficulties surrounding the continuum hypothesis. We show sufficient conditions for L(W) not to contain such a counterexample.  ...  On the other hand, large cardinals yield "sharps" for easily defined "inner-models" and consequently these inner-models cannot define a counterexample to the CH.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0168-0072(94)00031-w">doi:10.1016/0168-0072(94)00031-w</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4wtoduxe3vaw5cw6iz3h4tpjz4">fatcat:4wtoduxe3vaw5cw6iz3h4tpjz4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170928202015/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/6c9/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS8wMTY4MDA3Mjk0MDAwMzF3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f8/14/f8143ae3efa52330cda822a5ff60c8c0dd4c7aa3.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0168-0072(94)00031-w"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Page 6438 of Mathematical Reviews Vol. , Issue 96k [page]

<span title="">1996</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Kanamori (1-BOST; Boston, MA) 96k:03124 03E55 03E35 03E40 03ES0 Foreman, Matthew (1-CA3; Irvine, CA); Magidor, Menachem (IL-HEBR; Jerusalem) Large cardinals and definable counterexamples to the continuum  ...  The inner models had to do with accommodating large cardinals and evolved to the (Steel) core model up to a Woodin cardinal, the existence of several of which is currently the strongest hypothesis allowing  ... 
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Book Review: Aspects of constructibility

Menachem Magidor
<span title="1975-09-01">1975</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hjoli2j6qffdpaalkszryuidk4" style="color: black;">Bulletin of the American Mathematical Society</a> </i> &nbsp;
What made 1965 seem like a natural end to the Golden Age is, of course, P. J. Cohen's proof (1963) of the independence of the axiom of choice and of the continuum hypothesis.  ...  Besides, L satisfies the generalized continuum hypothesis, i.e. 2 Xot =N a +i for every infinite cardinal K«.  ...  This lemma (due to Gödel) plays a key role in subsequent chapters and in the present chapter, proving the generalized continuum hypothesis in L.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9904-1975-13860-2">doi:10.1090/s0002-9904-1975-13860-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/avigu72rjndfflrxfshjdcu7qe">fatcat:avigu72rjndfflrxfshjdcu7qe</a> </span>
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A GCH example of an ordinal graph with no infinite path

Jean A. Larson
<span title="1987-01-01">1987</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/w3g32txdvneltemssag5nwfxcy" style="color: black;">Transactions of the American Mathematical Society</a> </i> &nbsp;
In this paper, the Generalized Continuum Hypothesis is used to construct counterexamples for a cofinal set of ordinals of cardinality Nm, where m is a natural number at least two.  ...  It is hard to find nontrivial positive partition relations which hold for many ordinals in ordinary set theory, or even ordinary set theory with the additional assumption of the Generalized Continuum Hypothesis  ...  However, under the assumption of the Generalized Continuum Hypothesis, above the continuum, counterexamples start to appear, as indicated in the statement of the main theorem below. THEOREM.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9947-1987-0896028-6">doi:10.1090/s0002-9947-1987-0896028-6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wg52wborhnhwzfuoeltuj4nvdm">fatcat:wg52wborhnhwzfuoeltuj4nvdm</a> </span>
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A GCH Example of an Ordinal Graph with no Infinite Path

Jean A. Larson
<span title="">1987</span> <i title="JSTOR"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/w3g32txdvneltemssag5nwfxcy" style="color: black;">Transactions of the American Mathematical Society</a> </i> &nbsp;
In this paper, the Generalized Continuum Hypothesis is used to construct counterexamples for a cofinal set of ordinals of cardinality Nm, where m is a natural number at least two.  ...  It is hard to find nontrivial positive partition relations which hold for many ordinals in ordinary set theory, or even ordinary set theory with the additional assumption of the Generalized Continuum Hypothesis  ...  However, under the assumption of the Generalized Continuum Hypothesis, above the continuum, counterexamples start to appear, as indicated in the statement of the main theorem below. THEOREM.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2307/2000799">doi:10.2307/2000799</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gd6hrsjlmrdjnhwyzhtmm6ht4a">fatcat:gd6hrsjlmrdjnhwyzhtmm6ht4a</a> </span>
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The Vaught Conjecture: Do Uncountable Models Count?

John T. Baldwin
<span title="">2007</span> <i title="Duke University Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/un57kpou5jawnm7yjqoeumbno4" style="color: black;">Notre Dame Journal of Formal Logic</a> </i> &nbsp;
The following result is new. Theorem. If a first order theory is a counterexample to the Vaught conjecture then it has 2 ℵ 1 models of cardinality ℵ 1 .  ...  We give a model theoretic proof, replacing admissible set theory by the Lopez-Escobar theorem, of Makkai's theorem: Every counterexample to Vaught's conjecture has an uncountable model which realizes only  ...  To see this, rephrase 'extendible' as 'there exists an (∞, ω) map j from M properly into itself. (This relies on ≺ being (∞, ω)-submodel and the countability of M .)  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1305/ndjfl/1172787546">doi:10.1305/ndjfl/1172787546</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6cjuv5i3jvaffjdlodendbm6ne">fatcat:6cjuv5i3jvaffjdlodendbm6ne</a> </span>
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You Can Enter Cantor's Paradise [article]

Saharon Shelah
<span title="2001-02-07">2001</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
This paper is based on the talk given by the author after he received the International Bolyai Prize in Mathematics (on November 4, 2000 in Budapest, Hungary).  ...  Also mathematicians tend to conjecture either that whatever they cannot prove may fail, or whatever they cannot build counterexample to is true; and being unable to construct an intermediate cardinal between  ...  • We say that two sets A, B are equinumerous (or equivalent ) if there is a oneto-one and onto mapping from A onto B; • The Continuum Hypothesis, CH, is the following statement: every infinite set of reals  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0102056v1">arXiv:math/0102056v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/n2rwxuggrzgdrc267ekj5pwml4">fatcat:n2rwxuggrzgdrc267ekj5pwml4</a> </span>
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The Continuum Problem

John Stillwell
<span title="">2002</span> <i title="Informa UK Limited"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/6jb2mei4hzg4vdohokley4c5om" style="color: black;">The American mathematical monthly</a> </i> &nbsp;
Many models are now known to satisfy the continuum hypothesis, and it appears that no large cardinal axiom alone will contradict it.  ...  Nevertheless, there is another way in which large cardinal axioms can illuminate the continuum hypothesis, and this is the subject of our last section. DETERMINACY.  ... 
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On possible non-homeomorphic substructures of the real line

P. D. Welch
<span title="2002-02-12">2002</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/a64sw4kcsveajhudnssdwrwvze" style="color: black;">Proceedings of the American Mathematical Society</a> </i> &nbsp;
This requires large cardinals, and we obtain an exact consistency strength: Theorem 1.  ...  The reverse direction is a corollary to: We further consider the large cardinal consequences of the existence of a topological space with a proper substructure homeomorphic to Baire space.  ...  Jónsson cardinals and the continuum The following (Theorem 2 of the abstract) should be compared to Theorem 4 of [8] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9939-02-06385-2">doi:10.1090/s0002-9939-02-06385-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qf4jie2dujbizoxe4hklhgg2by">fatcat:qf4jie2dujbizoxe4hklhgg2by</a> </span>
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An Axiom for Nonseparable Borel Theory

William G. Fleissner
<span title="">1979</span> <i title="JSTOR"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/w3g32txdvneltemssag5nwfxcy" style="color: black;">Transactions of the American Mathematical Society</a> </i> &nbsp;
We show that this axiom is valid in the model constructed by collapsing a supercompact cardinal to <*>2 using Levy forcing.  ...  In this paper we discuss the consequences of the axiom of the title, among which are "yes" answers to both Kuratowski's and Stone's questions.  ...  The situation with respect to arrow 6 and large cardinals is different.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2307/1998696">doi:10.2307/1998696</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/erjodihlhvcbhb76nythtkji5m">fatcat:erjodihlhvcbhb76nythtkji5m</a> </span>
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An axiom for nonseparable Borel theory

William G. Fleissner
<span title="">1979</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/w3g32txdvneltemssag5nwfxcy" style="color: black;">Transactions of the American Mathematical Society</a> </i> &nbsp;
We show that this axiom is valid in the model constructed by collapsing a supercompact cardinal to <*>2 using Levy forcing.  ...  In this paper we discuss the consequences of the axiom of the title, among which are "yes" answers to both Kuratowski's and Stone's questions.  ...  The situation with respect to arrow 6 and large cardinals is different.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9947-1979-0531982-9">doi:10.1090/s0002-9947-1979-0531982-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/y36ehur4brea7nm7qie7ip5ray">fatcat:y36ehur4brea7nm7qie7ip5ray</a> </span>
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Page 03 of Mathematical Reviews Vol. , Issue 89M [page]

<span title="">1989</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
This paper gives a complete, self-contained proof of projective determinacy from a large cardinal hypothesis [the authors, Proc. Nat. Acad. Sci.  ...  Of three produced models of ZFC, one satisfies the continuum hypothesis; in another model, the constructed graph in addition does not contain any complete subgraph of size 4.  ... 
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Normal, not paracompact spaces

William G. Fleissner
<span title="1982-07-01">1982</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/iwegpttow5b4vhkhs3scejn5fi" style="color: black;">Bulletin of the American Mathematical Society</a> </i> &nbsp;
The work includes beautiful theorems, important counterexamples, and natural, unanswered questions.  ...  Let S w be the set of functions to cjj with domain {0, 1, 2, ...,«-1}. For a G S n we define N Q = {/GF: a C ƒ } = {fGF: /(O) = a(0), f{n -1) = a(n -1)}.  ...  [F2] ; Normal Moore spaces, continuum hypothesis and large cardinals, Proc. Nat. Acad. Sci. U. S. A. (to appear).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0273-0979-1982-15020-0">doi:10.1090/s0273-0979-1982-15020-0</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mlrkvqymivarfa5msyhh2om5pe">fatcat:mlrkvqymivarfa5msyhh2om5pe</a> </span>
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Page 6392 of Mathematical Reviews Vol. , Issue 91M [page]

<span title="">1991</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Summary: “Both the continuum hypothesis (CH) and Martin’s axiom (MA) allow inductive constructions to continue in cir- cumstances where the inductive hypothesis might otherwise fail.  ...  The application given here is to singular cardinals, the question being to what extent these will have cofinal sequences which are definable.  ... 
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The Radon-Nikodým Property Does Not Imply the Separable Complementation Property

Anatolij M. Plichko, David Yost
<span title="">2001</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/6ijda3dw7fgqbgpjopyzft3hhy" style="color: black;">Journal of Functional Analysis</a> </i> &nbsp;
There is a Banach space X enjoying the Radon-Nikody m Property and a separable subspace Y which is not contained in any complemented separable subspace of X. Academic Press  ...  ACKNOWLEDGMENT We are indebted to Ondr ej Kalenda for numerous helpful comments.  ...  Kernels of particular quotient maps from l 1 (:) onto l 2 (which exist if : has cardinality at least the continuum) were shown in [15, Section 4] to be counterexamples to another complementation problem  ... 
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