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Large cardinals and locally defined well-orders of the universe

David Asperó, Sy-David Friedman
<span title="">2009</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;
It is also possible to define variants of this construction which, in addition to forcing a locally defined well-order of the universe, preserve many of the n-huge cardinals from the ground model (for  ...  By forcing over a model of ZFC + GCH (above ℵ 0 ) with a class-sized partial order preserving this theory we produce a model in which there is a locally defined well-order of the universe; that is, one  ...  Acknowledgments The second author's research was supported by FWF Project P 19375-N18.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.apal.2008.07.006">doi:10.1016/j.apal.2008.07.006</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kicfi5q3ufhxzijzhfilueomym">fatcat:kicfi5q3ufhxzijzhfilueomym</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190412081035/https://core.ac.uk/download/pdf/82356670.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/9a/0f/9a0fdaf2273f3328ae1d3b7a429bd63d7313ce93.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.apal.2008.07.006"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Local Sentences and Mahlo Cardinals [article]

Olivier Finkel
<span title="2007-05-16">2007</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Ressayre who proved certain remarkable stretching theorems establishing the equivalence between the existence of finite models for these sentences and the existence of some infinite well ordered models  ...  Two of these stretching theorems were only proved under certain large cardinal axioms but the question of their exact (consistency) strength was left open in [O. Finkel and J.-P.  ...  In section 3 we prove the existence of a local sentence φ n which has some well ordered models of order type τ , for every infinite ordinal τ > ω which is not an n-Mahlo cardinal.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0705.2298v1">arXiv:0705.2298v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fko6nafcdrg6dkdlprdggvrjxa">fatcat:fko6nafcdrg6dkdlprdggvrjxa</a> </span>
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Set theory for category theory [article]

Michael A. Shulman
<span title="2008-10-07">2008</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets).  ...  In this expository paper we summarize and compare a number of such "set-theoretic foundations for category theory," and describe their implications for the everyday use of category theory.  ...  Just as the well-ordering of N justifies the usual sort of mathematical induction, the well-ordering of ordinals justifies definition and proof by transfinite induction.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0810.1279v2">arXiv:0810.1279v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2twv7sh2nbco7cgktblh2bs2tq">fatcat:2twv7sh2nbco7cgktblh2bs2tq</a> </span>
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Large cardinals and definable well-orders on the universe

Andrew D. Brooke-Taylor
<span title="">2009</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;
We use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals.  ...  This is achieved by coding using the principle , at a proper class of cardinals κ.  ...  This research was conducted at the Kurt Gödel Research Center for Mathematical Logic, The University of Vienna, with support from the Austrian Science Fund (FWF) project P16790-N04.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2178/jsl/1243948331">doi:10.2178/jsl/1243948331</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7v6m4y4djzhlhgyykdma7rnjme">fatcat:7v6m4y4djzhlhgyykdma7rnjme</a> </span>
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On Decidability Properties of Local Sentences

Olivier Finkel
<span title="">2005</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/uy5mv2ncw5eahkdx47hkrglxmm" style="color: black;">Electronical Notes in Theoretical Computer Science</a> </i> &nbsp;
Assuming the consistency of an inaccessible cardinal we prove that the set of local sentences having a model of order type ω2 is not determined by the axiomatic system ZF C + GCH, where GCH is the generalized  ...  The first one is a new decidability result in the case of local sentences whose function symbols are at most unary: one can decide, for every regular cardinal ωα (the α-th cardinal), whether a local sentence  ...  Finally sentences (3) and (6) are used to trivially define the functions h and k in other cases. We have seen that in a well ordered model M of Φ, P M 1 and P M 2 will be of order type ≤ ω 1 .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.entcs.2004.05.013">doi:10.1016/j.entcs.2004.05.013</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/52xyyuf5t5gyjdoamag2hppyny">fatcat:52xyyuf5t5gyjdoamag2hppyny</a> </span>
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On decidability properties of local sentences

Olivier Finkel
<span title="">2006</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
Next we prove that for all integers n, p 1, if n < p then the local theory of n , i.e. the set of local sentences having a model of order type n , is recursive in the local theory of p and also in the  ...  Assuming the consistency of an inaccessible cardinal we prove that the set of local sentences having a model of order type 2 is not determined by the axiomatic system ZFC + GCH, where GCH is the generalized  ...  Acknowledgements Thanks to the anonymous referee for useful comments on a preliminary version of this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.tcs.2006.08.004">doi:10.1016/j.tcs.2006.08.004</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/m2j2fqqlyjdsjmtlx3tqzojewa">fatcat:m2j2fqqlyjdsjmtlx3tqzojewa</a> </span>
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Extending the Language of Set Theory [article]

Dmytro Taranovsky
<span title="2016-12-19">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We discuss the problems of incompleteness and inexpressibility.  ...  We introduce almost self-referential formulas, use them to extend set theory, and relate their expressive power to that of infinitary logic. We discuss the nature of proper classes.  ...  Locally means without invoking large cardinals; definitions in V ω+ω are local, but definitions invoking the supremum of ordinals definable in first order set theory are not.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0504375v2">arXiv:math/0504375v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/slhhmaa6erhjjgyc5xp4lnnfcq">fatcat:slhhmaa6erhjjgyc5xp4lnnfcq</a> </span>
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On Decidability Properties of Local Sentences [article]

Olivier Finkel
<span title="2007-12-02">2007</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Assuming the consistency of an inaccessible cardinal we prove that the set of local sentences having a model of order type omega_2 is not determined by the axiomatic system ZFC + GCH, where GCH is the  ...  Local (first order) sentences, introduced by Ressayre, enjoy very nice decidability properties, following from some stretching theorems stating some remarkable links between the finite and the infinite  ...  Thanks to the anonymous referee for useful comments on a preliminary version of this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0712.0164v1">arXiv:0712.0164v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dfprkdsjxbbknfdrhdu5frkouy">fatcat:dfprkdsjxbbknfdrhdu5frkouy</a> </span>
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Page 615 of Mathematical Reviews Vol. , Issue 94b [page]

<span title="">1994</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;
The existence of a universal linear order in cardinality A is reduced to the existence of a universal model for any theory possessing the strict order property.  ...  At a singular cardinal yw, if uw is not a strong limit and is not a fixed point of the 8 function, then there is no universal linear order in uw.  ... 
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ABSOLUTENESS, TRUTH, AND QUOTIENTS [chapter]

Ilijas Farah
<span title="2013-12-10">2013</span> <i title="WORLD SCIENTIFIC"> Infinity and Truth </i> &nbsp;
of the third-order arithmetic.  ...  Once the existence of an infinite set is accepted, the axioms of set theory imply the existence of a transfinite hierarchy of larger and larger orders of infinity.  ...  I would like to thank the organizers of both meetings.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1142/9789814571043_0001">doi:10.1142/9789814571043_0001</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/izq2d3pfh5dibay6bs5u63eniu">fatcat:izq2d3pfh5dibay6bs5u63eniu</a> </span>
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Large Cardinals and Topology: a Short Retrospective and Some New Results

S. G. Da Silva
<span title="2007-09-26">2007</span> <i title="Oxford University Press (OUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/mwzmwwmscfe4vnzrjlt5itd2ju" style="color: black;">Logic Journal of the IGPL</a> </i> &nbsp;
implies the existence of inner models with measurable cardinals.  ...  The author's purpose is to present some applications of large cardinals in general topology, pointing out that there are several topological problems that cannot be settled without dealing with inaccessible  ...  Acknowledgements The author would like to thank the referee for his (or her) careful reading of the paper and for providing several useful comments and suggestions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1093/jigpal/jzm052">doi:10.1093/jigpal/jzm052</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6edqvzc53fbzjmgz6kig7qn5qe">fatcat:6edqvzc53fbzjmgz6kig7qn5qe</a> </span>
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Consistency Results in Topology and Homotopy Theory

Jaykov Foukzon
<span title="">2015</span> <i title="Science Publishing Group"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wm7zsdl57val5h4reftbt6fk7y" style="color: black;">Pure and Applied Mathematics Journal</a> </i> &nbsp;
Main results is: (1) let κ be an inaccessible cardinal and H k is a set of all sets having hereditary size less then κ, then Con(ZFC + (V = H k )), (2) there is a Lindelöf T 3 indestructible space of pseudocharacter  ...  ≤N 1 and size N 2 in .  ...  Introduction A note on the term large cardinal is in order. A cardinal number is 'large' if the assumption of its existence, when added to the axioms of ZFC, proves the consistency of ZFC.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.11648/j.pamj.s.2015040101.11">doi:10.11648/j.pamj.s.2015040101.11</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fa7berac2narzf4jx5pzlnvehe">fatcat:fa7berac2narzf4jx5pzlnvehe</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170706125103/http://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2015040101.11.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/d0/63/d0632336d562ae1b8882909acd5b8b9365427c24.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.11648/j.pamj.s.2015040101.11"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Epireflections and supercompact cardinals

Joan Bagaria, Carles Casacuberta, Adrian R.D. Mathias
<span title="">2009</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/74pjbehpfraq5ogj3loetdwp7m" style="color: black;">Journal of Pure and Applied Algebra</a> </i> &nbsp;
In other words, if L is a localization functor on a balanced accessible category such that the unit morphism X → LX is an epimorphism for all X and the class of L-local objects is defined by an absolute  ...  formula, then the existence of a sufficiently large supercompact cardinal implies that L is a localization with respect to some set of morphisms.  ...  Locally presentable categories are complete, well-powered and co-well-powered.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jpaa.2008.11.046">doi:10.1016/j.jpaa.2008.11.046</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/o74iqjknrzclzl4bai5cosx66y">fatcat:o74iqjknrzclzl4bai5cosx66y</a> </span>
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Ineffability within the Limits of Abstraction Alone [chapter]

Stewart Shapiro, Gabriel Uzquiano
<span title="2016-12-08">2016</span> <i title="Oxford University Press"> Abstractionism </i> &nbsp;
The gold standard would be a theory as rich and useful as the dominant one, ZFC, Zermelo-Fraenkel set theory with choice, perhaps augmented with large cardinal principles.  ...  The standard exemplar, of course, is Hume's principle: where, as usual, F ≈ G is an abbreviation of the second-order statement that there is a one-one correspondence from F onto G.  ...  Acknowledgements We are grateful to audiences at the final Abstraction Workshop at Arché (Status Belli) and at the Mathematical Methods in Philosophy in Banff, where earlier versions of this paper were  ... 
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Implications of large-cardinal principles in homotopical localization

Carles Casacuberta, Dirk Scevenels, Jeffrey H. Smith
<span title="">2005</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/37jjomjmvfhrzf3di2gjnrrfuu" style="color: black;">Advances in Mathematics</a> </i> &nbsp;
We prove that both questions have an affirmative answer assuming the validity of a suitable large-cardinal axiom from set theory (Vopěnka's principle).  ...  The existence of arbitrary cohomological localizations on the homotopy category of spaces has remained unproved since Bousfield settled the same problem for homology theories in the decade of 1970.  ...  Hence, as explained in [20, 2.2] , the homotopy type of a homotopically small simplicial set is well defined within the universe U in which we agreed to work.  ... 
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