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Indestructibility, instances of strong compactness, and level by level inequivalence

Arthur W. Apter
<span title="2010-07-20">2010</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;
supercompact, and every measurable cardinal δ < κ which is not a limit of measurable cardinals is δ + strongly compact.  ...  The first of these contains a supercompact cardinal κ and is such that no cardinal δ > κ is measurable, κ's supercompactness is indestructible under κ-directed closed, (κ + , ∞)distributive forcing, and  ...  In addition, the following hold in V P : 1. κ is indestructibly supercompact. 2 . κ is the least strongly compact cardinal. 3 .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00153-010-0200-0">doi:10.1007/s00153-010-0200-0</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4kdqvvwrp5ggtctaxtw2f3yssi">fatcat:4kdqvvwrp5ggtctaxtw2f3yssi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170922234300/http://faculty.baruch.cuny.edu/aapter/papers/ind08c.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/49/dc/49dcffe24a655b11617b39a9238d2428de8edbe3.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00153-010-0200-0"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

HOD-supercompactness, Indestructibility, and Level by Level Equivalence

Arthur W. Apter, Shoshana Friedman
<span title="">2014</span> <i title="Institute of Mathematics, Polish Academy of Sciences"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/m4uvgcbkwrcwrphi44yha6kndy" style="color: black;">Bulletin of the Polish Academy of Sciences Mathematics</a> </i> &nbsp;
, the strongly compact and supercompact cardinals coincide except at measurable limit points, and level by level equivalence between strong compactness and supercompactness holds above κ 0 but fails below  ...  Additionally, we get the property of being supercompact but not hod-supercompact at the least supercompact cardinal, in a model where level by level equivalence between strong compactness and supercompactness  ...  This is since otherwise, some cardinal γ < κ 0 is supercompact up to a strong cardinal and hence is fully supercompact.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4064/ba62-3-1">doi:10.4064/ba62-3-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/no44tcqvlfbafdg74t4uxspqbu">fatcat:no44tcqvlfbafdg74t4uxspqbu</a> </span>
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A note on strong compactness and resurrectibility

Arthur W. Apter
<span title="">2000</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 construct a model containing a proper class of strongly compact cardinals in which no strongly compact cardinal κ is κ + supercompact and in which every strongly compact cardinal has its strong compactness  ...  For instance, in the list of open questions at the end of [1] , it is asked whether the first α strongly compact cardinals can be non-supercompact and still exhibit some sort of indestructibility properties  ...  As before, if we start the definition of P 1 by adding a Cohen real, since no measurable limit of strongly compact cardinals λ ∈ V 1 will be λ + supercompact, we can use the results of [8] , [9] , and  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4064/fm-165-3-258-290">doi:10.4064/fm-165-3-258-290</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/72uoafjf3zbobnitblsso23b4a">fatcat:72uoafjf3zbobnitblsso23b4a</a> </span>
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Indestructibility and the level-by-level agreement between strong compactness and supercompactness [article]

Arthur W. Apter and Joel David Hamkins (CUNY and Carnegie Mellon University)
<span title="2001-02-10">2001</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Can a supercompact cardinal kappa be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness?  ...  it only on measure one sets, then yes, it can.  ...  The argument also works for singular η of arbitrary cofinality above κ, the basic point being that if γ is <η-strongly compact and η is singular with cofinality at least γ, then γ is η-strongly compact  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0102086v1">arXiv:math/0102086v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kp56uovh25bbrk5ed7v5ipbw2m">fatcat:kp56uovh25bbrk5ed7v5ipbw2m</a> </span>
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Indestructibility and destructible measurable cardinals

Arthur W. Apter
<span title="2015-12-12">2015</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;
On the other hand, under the same hypotheses, A 2 = {δ < κ | δ is measurable, δ is not a limit of measurable cardinals, δ is not δ + strongly compact, and δ's measurability is indestructible when forcing  ...  It then follows that A 1 = {δ < κ | δ is measurable, δ is not a limit of measurable cardinals, δ is not δ + strongly compact, and δ's measurability is destructible when forcing with partial orderings having  ...  supercompact and is also the least strongly compact cardinal + Any measurable cardinal δ < κ which is not a limit of measurable cardinals is <λ δ strongly compact and has its <λ δ strong compactness (  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00153-015-0470-7">doi:10.1007/s00153-015-0470-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/f5pkzryxkngwhoe3mzbkvf6scu">fatcat:f5pkzryxkngwhoe3mzbkvf6scu</a> </span>
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Some applications of Sargsyan's equiconsistency method

Arthur W. Apter
<span title="">2012</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 then show how these and additional techniques due to Sargsyan may be employed to establish an equiconsistency for a related indestructibility theorem for strongness.  ...  We apply techniques due to Sargsyan to reduce the consistency strength of the assumptions used to establish an indestructibility theorem for supercompactness.  ...  The author wishes to thank the referee for helpful comments, suggestions, and corrections which have been incorporated into the current version of the paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4064/fm216-3-2">doi:10.4064/fm216-3-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qn4gv7zqrzgondlkdnhllagtb4">fatcat:qn4gv7zqrzgondlkdnhllagtb4</a> </span>
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Indestructibility, strong compactness, and level by level equivalence

Arthur W. Apter
<span title="">2009</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 show the relative consistency of the existence of two strongly compact cardinals κ1 and κ2 which exhibit indestructibility properties for their strong compactness, together with level by level equivalence  ...  In the model constructed, κ1's strong compactness is indestructible under arbitrary κ1-directed closed forcing, κ1 is a limit of measurable cardinals, κ2's strong compactness is indestructible under κ2  ...  In addition, the author wishes to thank the referee for helpful comments, suggestions, and corrections which have been incorporated into the current version of the paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4064/fm204-2-2">doi:10.4064/fm204-2-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rqya7xmehjc3foggdsgxwldpei">fatcat:rqya7xmehjc3foggdsgxwldpei</a> </span>
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Indestructible Strong Compactness and Level by Level Equivalence with No Large Cardinal Restrictions

Arthur W. Apter
<span title="">2015</span> <i title="Institute of Mathematics, Polish Academy of Sciences"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/m4uvgcbkwrcwrphi44yha6kndy" style="color: black;">Bulletin of the Polish Academy of Sciences Mathematics</a> </i> &nbsp;
We construct a model for the level by level equivalence between strong compactness and supercompactness with an arbitrary large cardinal structure in which the least supercompact cardinal κ has its strong  ...  . † Keywords: Supercompact cardinal, strongly compact cardinal, indestructibility, Gitik iteration, Magidor iteration of Prikry forcing, level by level equivalence between strong compactness and supercompactness  ...  with the least supercompact cardinal κ having its strong compactness indestructible under any κ-directed closed forcing notion.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4064/ba8014-12-2015">doi:10.4064/ba8014-12-2015</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tnnqatc4p5ct7bveqqz4cf7y7e">fatcat:tnnqatc4p5ct7bveqqz4cf7y7e</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170809013505/http://faculty.baruch.cuny.edu/aapter/papers/ind14a.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/5b/48/5b48fdf62c8b55e8ec3b3355fc2f99832553d5df.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4064/ba8014-12-2015"> <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>

Exactly controlling the non-supercompact strongly compact cardinals [article]

Arthur W. Apter, Joel David Hamkins
<span title="2003-01-03">2003</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals  ...  We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants.  ...  In one of the models constructed, the least strongly compact cardinal is also the least measurable cardinal, yet indestructible. 4 .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0301016v1">arXiv:math/0301016v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/znxl2yyz4vcpjjzwbmyjvysrfm">fatcat:znxl2yyz4vcpjjzwbmyjvysrfm</a> </span>
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Patterns of Compact Cardinals [article]

Arthur W. Apter
<span title="1999-03-01">1999</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
of the theorem the consistency of the least measurable limit of supercompact cardinals being the same as the least measurable limit of non-supercompact strongly compact cardinals and the consistency of  ...  the least supercompact cardinal being a limit of strongly compact cardinals.  ...  to the consistency of a strongly compact cardinal, for the least strongly compact cardinal to be the least measurable cardinal (in which case, it is not the least supercompact cardinal), but that it was  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/9903010v1">arXiv:math/9903010v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/r2vfksfq4vdmbkwm3nleqwdjnm">fatcat:r2vfksfq4vdmbkwm3nleqwdjnm</a> </span>
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Exactly controlling the non-supercompact strongly compact cardinals

Arthur W. Apter, Joel David Hamkins
<span title="">2003</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/4l7nckgxmbcgvj5vxsioq6qwyq" style="color: black;">Journal of Symbolic Logic (JSL)</a> </i> &nbsp;
Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals  ...  We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants.  ...  In one of the models constructed, the least strongly compact cardinal is also the least measurable cardinal, yet indestructible. 4 .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2178/jsl/1052669070">doi:10.2178/jsl/1052669070</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/szzk73eqxvbllne3j6v2notbpy">fatcat:szzk73eqxvbllne3j6v2notbpy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170829052026/http://logic.amu.edu.pl/images/b/bc/Apterhamkins.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/73/a5/73a5004e0395b5f3286b19ca4235aee44e7cb06d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2178/jsl/1052669070"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Patterns of compact cardinals

Arthur W. Apter
<span title="">1997</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;
of the theorem the consistency of the least measurable limit of supercompact cardinals being the same as the least measurable limit of nonsupercompact strongly compact cardinals and the consistency of  ...  the least supercompact cardinal being a limit of strongly compact cardinals.  ...  In addition, the author wishes to thank the referee, both for promptly reviewing the paper and for making helpful comments and suggestions which improved the presentation of the material contained herein  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0168-0072(97)80001-2">doi:10.1016/s0168-0072(97)80001-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2loyw5lpsveincxeajvgzhjdhe">fatcat:2loyw5lpsveincxeajvgzhjdhe</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170927072326/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/4e1/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDE2ODAwNzI5NzgwMDAxMg%3D%3D.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/84/33/8433cec94cc136a115b5ef39e022745e0f68378f.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0168-0072(97)80001-2"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Mixed Levels of Indestructibility

Arthur W. Apter
<span title="">2015</span> <i title="Institute of Mathematics, Polish Academy of Sciences"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/m4uvgcbkwrcwrphi44yha6kndy" style="color: black;">Bulletin of the Polish Academy of Sciences Mathematics</a> </i> &nbsp;
Starting from a supercompact cardinal κ, we force and construct a model in which κ is both the least strongly compact and least supercompact cardinal and κ exhibits mixed levels of indestructibility.  ...  Specifically, κ's strong compactness, but not its supercompactness, is indestructible under any κ-directed closed forcing which also adds a Cohen subset of κ.  ...  There is then a partial ordering P ⊆ V such that V P "κ is both supercompact and the least strongly compact cardinal". For any Q ∈ V P which is κ-directed closed, V P * Q "κ is strongly compact".  ... 
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Inner models with large cardinal features usually obtained by forcing

Arthur W. Apter, Victoria Gitman, Joel David Hamkins
<span title="2011-12-22">2011</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;
If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact.  ...  If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly  ...  By the arguments of [Mag76] , the cardinal κ becomes both the least strongly compact and the least measurable cardinal in V P δ .  ... 
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Page 7296 of Mathematical Reviews Vol. , Issue 2003j [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;
«, and «2, both the first two strongly compact and the first two measurable cardinals and such that K\’s strong compactness and «2’s measurability are indestructible for «\-directed and « -directed closed  ...  supercompact cardinals, the strongly compact and supercompact cardinals coincide (except at measurable limits), and all supercompact cardinals are indestruc- tible, as are all strongly compact cardinals  ... 
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