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Page 8923 of Mathematical Reviews Vol. , Issue 2003m [page]

<span title="">2003</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;
(1-CA; Berkeley, CA) The consistency strength of successive cardinals with the tree property.  ...  “Cardinals much larger than weakly compacts are indeed neces- sary for these patterns of cardinals with the tree property to exist.  ... 
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Definable tree property for successors of cardinals [article]

Ali Sadegh Daghighi, Massoud Pourmahdian
<span title="2015-11-21">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
It is proved that the consistency strength of having definable tree property for successors of all regular cardinals is the consistency strength of having proper class many small large cardinals which  ...  Also the consistency strength of definable tree property for successor of a singular cardinal is reduced to the existence of a supercompact cardinal and a measurable above it.  ...  The question regarding the consistency and consistency strength of usual tree property for successors of all regular cardinals is still open.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1511.04598v2">arXiv:1511.04598v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/j6oes7tgkjbxfk7ioo2gbztab4">fatcat:j6oes7tgkjbxfk7ioo2gbztab4</a> </span>
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The definable tree property for successors of cardinals

Ali Sadegh Daghighi, Massoud Pourmahdian
<span title="2016-06-09">2016</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3egevcnserbbni4tcigsztdici" style="color: black;">Archive for Mathematical Logic</a> </i> &nbsp;
Strengthening a result of Amir Leshem [7], we prove that the consistency strength of holding GCH together with definable tree property for all successors of regular cardinals is precisely equal to the  ...  consistency strength of existence of proper class many Π 1 1 -reflecting cardinals.  ...  The question regarding the consistency and consistency strength of usual tree property for successors of all regular cardinals is still open.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00153-016-0494-7">doi:10.1007/s00153-016-0494-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ydm7lcv74zaoljy2xknlackcsa">fatcat:ydm7lcv74zaoljy2xknlackcsa</a> </span>
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Fragility and indestructibility II

Spencer Unger
<span title="">2015</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;
Techniques for preserving the tree property are central to a growing literature of consistency results obtaining the tree property as successive regular cardinals (see for example [11, 1, 3, 15, 20]).  ...  It follows that the tree property at ω 2 is consistent with 2 ω > ω 2 . The preservation lemma was Lemma 0.1. Let τ, η be cardinals with η regular and 2 τ ≥ η. Let P be τ + -cc and R be τ + -closed.  ...  Variants of the following two claims appear in the literature on successive cardinals with the tree property.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.apal.2015.06.002">doi:10.1016/j.apal.2015.06.002</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/suxxtdrjdvfxjiq3t46oany6fy">fatcat:suxxtdrjdvfxjiq3t46oany6fy</a> </span>
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Large Cardinals with Forcing [chapter]

Akihiro Kanamori
<span title="">2012</span> <i title="Elsevier"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3htcvojut5de5ct6zf42o424vi" style="color: black;">Handbook of the History of Logic</a> </i> &nbsp;
2 having the tree property is equi-consistent to having a weakly compact cardinal.  ...  Finally, for a cardinal κ, a κ-tree is a tree of height κ each of whose levels has cardinality less than κ, and κ has the tree property iff every κ-tree has a cofinal branch.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/b978-0-444-51621-3.50004-9">doi:10.1016/b978-0-444-51621-3.50004-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rxofgx3z6nelvlyhmdybzlox6e">fatcat:rxofgx3z6nelvlyhmdybzlox6e</a> </span>
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Page 3865 of Mathematical Reviews Vol. , Issue 2000f [page]

<span title="">2000</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;
As an application we obtain that the difference property of the class Y is consistent with ZFC.”  ...  . (2) Consistency of a measurable cardinal implies the consis- tency of Ref(N3,N2,No) + Smr(X3, 2,8.) + Snr(X3, &), No). (3) Consistency of a supercompact cardinal with a mea- surable above implies the  ... 
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Kunen and set theory

Akihiro Kanamori
<span title="">2011</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hp7o5f6pmrg57k2m46s4syavf4" style="color: black;">Topology and its Applications</a> </i> &nbsp;
With the emergence of the now basic ultraproduct construction in model theory, Dana Scott in 1961 took an ultrapower of the entire set-theoretic universe V to establish that having a measurable cardinal  ...  He expresses his gratitude for the productive conditions and support of the kolleg. 2 See Jech [33] for the set-theoretic concepts and results, and Kanamori [38] for those involving large cardinals. 0166  ...  By his work [45] with iterated ultrapowers, the consistency strength of having such an ideal was stronger than having a measurable cardinal.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.topol.2011.08.002">doi:10.1016/j.topol.2011.08.002</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/awclymp5rzg23ovxzj6a5hsy7e">fatcat:awclymp5rzg23ovxzj6a5hsy7e</a> </span>
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Large Cardinal Properties of Small Cardinals [chapter]

James Cummings
<span title="">1998</span> <i title="Springer Netherlands"> Set Theory </i> &nbsp;
Fact 5. 6 ( 6 Foreman and Magidor) If two successive cardinals have the tree property, there is an inner model with a strong cardinal.  ...  If it is consistent that there are ! supercompact cardinals, it is consistent that @ n has the tree property for every n with 2 n < !.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-94-015-8988-8_2">doi:10.1007/978-94-015-8988-8_2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/q2a7cr4aendnvosxt65p4uczky">fatcat:q2a7cr4aendnvosxt65p4uczky</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20120712144855/http://www.math.cmu.edu/users/jcumming/papers/barc.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/6f/db/6fdb9c80d5d17f525c18e99252a167f4596dcb22.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-94-015-8988-8_2"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

The consistency strength of the tree property at the double successor of a measurable cardina

Natasha Dobrinen, Sy-David Friedman
<span title="">2010</span> <i title="Institute of Mathematics, Polish Academy of Sciences"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yeobluctzze7rfwekxs7sx2ayy" style="color: black;">Fundamenta Mathematicae</a> </i> &nbsp;
We also find upper and lower bounds for the consistency strength of there being no special Aronszajn trees at the double successor of a measurable cardinal.  ...  model in which there is a proper class of measurable cardinals, and in which the tree property holds at the double successor of each strongly inaccessible cardinal.  ...  The first author would like to thank FWF grant P16790 -N04 for support of this research. The second author would like to thank FWF grants P16790 -N04 and P19375 -N18 for support of this research.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4064/fm208-2-2">doi:10.4064/fm208-2-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/sl4nlylh4jdupkmnpvpadk7o5q">fatcat:sl4nlylh4jdupkmnpvpadk7o5q</a> </span>
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Laver and set theory

Akihiro Kanamori
<span title="2016-01-18">2016</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3egevcnserbbni4tcigsztdici" style="color: black;">Archive for Mathematical Logic</a> </i> &nbsp;
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.  ...  Doctoral students of Richard Laver Stephen Grantham, An analysis of Galvin In addition to having these doctoral students at Boulder, Laver was on the thesis committees of, among many:  ...  In [69] Laver established the consistency of a substantial version of his saturation property holding for a κ-ideal with κ a successor cardinal, thereby establishing the consistency of a partition property  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00153-015-0462-7">doi:10.1007/s00153-015-0462-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5ziwvh27arbprhsdlmku4f2npe">fatcat:5ziwvh27arbprhsdlmku4f2npe</a> </span>
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Strong tree Properties for two successive cardinals [article]

Laura Fontanella
<span title="2011-12-14">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove that if there is a model of with two supercompact cardinals, then there is a model of where simultaneously (_2, μ)-ITP and (_3, μ')-ITP hold, for all μ≥_2 and μ'≥_3.  ...  An inaccessible cardinal κ is supercompact when (κ, λ)-ITP holds for all λ≥κ.  ...  Let's discuss a naive attempt to build a model of the super tree property for two successive cardinals ℵ n , ℵ n+1 (with n ≥ 2).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1110.6736v5">arXiv:1110.6736v5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/iiaryjhxybgn7d3cmct46ji44u">fatcat:iiaryjhxybgn7d3cmct46ji44u</a> </span>
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Large cardinals and their effect on the continuum function on regular cardinals

Radek Honzík
<span title="2016-08-08">2016</span> <i title="Charles University in Prague, Karolinum Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lq4kh6tyqrc75dbkot4lhm42ka" style="color: black;">Acta Universitatis Carolinae: Philosophica et Historica</a> </i> &nbsp;
The paper concludes with a result from the opposite end of the spectrum: namely, how to kill all large cardinals in the universe.  ...  In this survey paper, we will summarise some of the more and less known results on the generalisation of the Easton theorem in the context of large cardinals.  ...  Acknowledgments The author was supported by GAČR project I 1921-N25.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.14712/24647055.2016.9">doi:10.14712/24647055.2016.9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/v6jtg6naezhwrpi5cobyubbuei">fatcat:v6jtg6naezhwrpi5cobyubbuei</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180721174030/http://www.karolinum.cz/ink2_stat/dload.jsp?prezMat=109692" 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/47/d9/47d92482ac09a6465ae04710a5b2b885fba5b5bd.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.14712/24647055.2016.9"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Projective determinacy

D. A. Martin, J. R. Steel
<span title="1988-09-01">1988</span> <i title="Proceedings of the National Academy of Sciences"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nvtuoas5pbdsllkntnhizy4f4q" style="color: black;">Proceedings of the National Academy of Sciences of the United States of America</a> </i> &nbsp;
The ZFC (Zermelo-Fraenkel, with the axiom of choice) axi-Abbreviations: ZFC, Zermelo-Fraenkel, with the axiom of choice; PD, projective determinacy; AD, axiom of determinacy. 6582 The publication costs  ...  It is shown that projective determinacy follows from large cardinal axioms weaker than the assertion that supercompact cardinals exist.  ...  For 0 < a ' c, an iteration tree on V of length a is a system (<, (Mk k < a), (Ek k + 1 < a), (Pk k + 1 < a)) with the following properties: (i) (a, <) is a tree ordering with 0 <-least and with n < m  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1073/pnas.85.18.6582">doi:10.1073/pnas.85.18.6582</a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pubmed/16593978">pmid:16593978</a> <a target="_blank" rel="external noopener" href="https://pubmed.ncbi.nlm.nih.gov/PMC282021/">pmcid:PMC282021</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nd4x5r73zzdmbjwspv4u3a2p2e">fatcat:nd4x5r73zzdmbjwspv4u3a2p2e</a> </span>
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A general tool for consistency results related to I1 [article]

Vincenzo Dimonte, Liuzhen Wu
<span title="2015-10-12">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper we provide a general tool to prove the consistency of I1(λ) with various combinatorial properties at λ typical at settings with 2^λ>λ^+, that does not need a profound knowledge of the forcing  ...  Examples of such properties are the first failure of GCH, a very good scale and the negation of the approachability property, or the tree property at λ^+ and λ^++.  ...  T is a tree, with a trunk p max(g) , consisting of • -increasing sequences.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1510.03287v1">arXiv:1510.03287v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/d44lvu4hvzg2nfccqtuuzilo4y">fatcat:d44lvu4hvzg2nfccqtuuzilo4y</a> </span>
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A general tool for consistency results related to I1

Vincenzo Dimonte, Liuzhen Wu
<span title="2015-12-18">2015</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/p66xx56fyjhgzmbhbcacs75jze" style="color: black;">European Journal of Mathematics</a> </i> &nbsp;
In this paper we provide a general tool to prove the consistency of I1(λ) with various combinatorial properties at λ typical at settings with 2 λ > λ + , that does not need a profound knowledge of the  ...  Examples of such properties are the first failure of GCH, a very good scale and the negation of the approachability property, or the tree property at λ + and λ ++ .  ...  While this research heavily involves large cardinals, their role has almost always been giving consistency strength to a certain scenery, but they rarely appear directly with the desired combinatorial  ... 
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