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A Mid Version of Hamkins' Maximality Principle [article]

Rahman Mohammadpour
<span title="2015-11-28">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Moreover, we formulate an intermediate maximality principle, which is shown here to be equiconsistent with the existence of a weakly compact cardinal κ such that V_κ≺ V.  ...  We present new, streamlined proofs of certain maximality principles studied by Hamkins and Woodin.  ...  MP(R) and show that it is equiconsistent with the existence of a weakly compact cardinal κ such that V κ ≺ V .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1506.03902v3">arXiv:1506.03902v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ojq4iwhwp5cwlaqavkfj5df4hu">fatcat:ojq4iwhwp5cwlaqavkfj5df4hu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200903183247/https://arxiv.org/pdf/1506.03902v3.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/e1/b4/e1b4d5b57248a6064d3ba02dd1f69b47dbcfc57b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1506.03902v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

The Necessary Maximality Principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal [article]

Joel David Hamkins, W. Hugh Woodin
<span title="2004-03-09">2004</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that this principle is equiconsistent with the existence of a weakly compact cardinal.  ...  The Necessary Maximality Principle for c.c.c. forcing asserts that any statement about a real in a c.c.c. extension that could become true in a further c.c.c. extension and remain true in all subsequent  ...  Main Theorem 3 The principle 2mpccc(R) is equiconsistent over zfc with the existence of a weakly compact cardinal.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0403165v1">arXiv:math/0403165v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bhe4iyai6vbovoy4fkwwyq47ju">fatcat:bhe4iyai6vbovoy4fkwwyq47ju</a> </span>
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A simple maximality principle [article]

Joel David Hamkins
<span title="2000-09-28">2000</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in phi, is equiconsistent with the scheme asserting that V_delta is an elementary substructure of V for an  ...  inaccessible cardinal delta, which in turn is equiconsistent with the scheme asserting that ORD is Mahlo.  ...  One can modify the assumption that ord is ineffable by building the core model up to a weakly compact cardinal κ.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0009240v1">arXiv:math/0009240v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/objpfd7exrg2xbm2x2ndpae6jq">fatcat:objpfd7exrg2xbm2x2ndpae6jq</a> </span>
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Strongly uplifting cardinals and the boldface resurrection axioms [article]

Joel David Hamkins, Thomas A. Johnstone
<span title="2014-10-30">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.  ...  We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost hugely unfoldable cardinals, and we show  ...  Theorem 19 therefore illustrates how the equiconsistency established in [HJ14] between the uplifting cardinals and the resurrection axioms generalizes to the boldface context, with the strongly uplifting  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1403.2788v2">arXiv:1403.2788v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/uxbpnmqeo5atrco52gumtmdpl4">fatcat:uxbpnmqeo5atrco52gumtmdpl4</a> </span>
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Closed maximality principles: implications, separations and combinations

Gunter Fuchs
<span title="">2008</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;
In contrast to this, it is equiconsistent with ZFC that the maximality principle for directed-closed forcings without any parameters holds at every regular cardinal.  ...  These principles come in many variants, depending on the parameters which are allowed, I shall write MPΓ (A) for the maximality principle for forcings in Γ, with parameters from A.  ...  So one arrives at the following questions: Let's call a weakly compact cardinal κ that remains weakly compact in any forcing extension by a <κ-closed partial order an indestructibly weakly compact cardinal  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2178/jsl/1208358754">doi:10.2178/jsl/1208358754</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zpwoiahaejgubf75jrqapmztxi">fatcat:zpwoiahaejgubf75jrqapmztxi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20100216100451/http://www.math.csi.cuny.edu:80/~fuchs/MPclosed.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/99/27/9927ba7425e49167e5fb736e3c8eb0fcdd7fd450.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2178/jsl/1208358754"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Page 21 of Mathematical Reviews Vol. , Issue 85a [page]

<span title="">1985</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;
For the difficult direction, (1) implies (2), the least inaccessible cardinal « in L is delicately collapsed by forcing to w; as follows: A sequence a reals (ag:a< - . adjoined with the properties: (a)  ...  It is that if |A| < ¢ and |A| < c, where A C A(A) \w, then there is a T C A(A) \w such that (a) || = ¢, (b) every member of [ is A-incomparable with every member of A, (c) any two distinct members of [  ... 
<span class="external-identifiers"> </span>
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Consistency Strengths of Modified Maximality Principles [article]

George Leibman
<span title="2004-06-03">2004</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The Maximality Principle MP is a scheme which states that if a sentence of the language of ZFC is true in some forcing extension V^P, and remains true in any further forcing extension of V^P, then it is  ...  A modified maximality principle MP_Gamma arises when considering forcing with a particular class Gamma of forcing notions.  ...  the compactness theorem to find a model of the full maximality principle.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0406063v1">arXiv:math/0406063v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nnysiq7k2rf2zfzzuv5q6o2hhu">fatcat:nnysiq7k2rf2zfzzuv5q6o2hhu</a> </span>
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Combined Maximality Principles up to large cardinals

Gunter Fuchs
<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;
The motivation for this paper is the following: In [4] I showed that it is inconsistent with ZFC that the Maximality Principle for directed closed forcings holds at unboundedly many regular cardinals κ  ...  (even only allowing κ itself as a parameter in the Maximality Principle for &lt;κ-closed forcings each time).  ...  The Maximality Principle for forcings in Γ with parameters in P , abbreviated as MP Γ (P ) is the scheme of formulae expressing that every formula with parameters in P that is Γ-forceably necessary is  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2178/jsl/1245158097">doi:10.2178/jsl/1245158097</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ucz44d5hx5aphmesbujedvxzwu">fatcat:ucz44d5hx5aphmesbujedvxzwu</a> </span>
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Resurrection axioms and uplifting cardinals [article]

Joel David Hamkins, Thomas A. Johnstone
<span title="2014-02-26">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent  ...  over ZFC with the existence of an uplifting cardinal.  ...  Essentially the same argument shows that c is weakly α-inaccessible for every α < c-so it is weakly hyper-inaccessibleand it is a limit of such cardinals, and so on.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1307.3602v2">arXiv:1307.3602v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cphqpohrsndg5dqwbjy5febgtu">fatcat:cphqpohrsndg5dqwbjy5febgtu</a> </span>
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Combining Resurrection and Maximality [article]

Kaethe Minden
<span title="2020-12-16">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
It is shown that the resurrection axiom and the maximality principle may be consistently combined for various iterable forcing classes.  ...  The extent to which resurrection and maximality overlap is explored via the local maximality principle.  ...  The local maximality principle is equiconsistent with the existence of a locally uplifting cardinal, using the same method as with the proof of the maximality principle but with some care in relativizing  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2006.03655v2">arXiv:2006.03655v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gfwte3cgpbgdlbk5zxqewzh4a4">fatcat:gfwte3cgpbgdlbk5zxqewzh4a4</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;
Mitchell resolved the problem by proving Fact 5.2 (Mitchell 31 ]) The following are equiconsistent. 1. There exists a weakly compact cardinal. 2. @ 2 has the tree property.  ...  It is known that the weakly compact cardinals are exactly those inaccessible cardinals which have the tree property.  ... 
<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>
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Resurrection axioms and uplifting cardinals

Joel David Hamkins, Thomas A. Johnstone
<span title="2014-02-25">2014</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;
We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent  ...  over ZFC with the existence of an uplifting cardinal.  ...  Essentially the same argument shows that c is weakly α-inaccessible for every α < c-so it is weakly hyperinaccessible-and it is a limit of such cardinals, and so on.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00153-014-0374-y">doi:10.1007/s00153-014-0374-y</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7a6cxh25jnhnphflkc547ok6wi">fatcat:7a6cxh25jnhnphflkc547ok6wi</a> </span>
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Equiconsistencies at subcompact cardinals

Itay Neeman, John Steel
<span title="2015-12-22">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;
As a corollary we also see that assuming the existence of a Woodin cardinal δ so that SBH δ holds, the Proper Forcing Axiom implies the existence of a class inner model with a Π 2 1 subcompact cardinal  ...  A thread through a coherent sequence is a club C ⊆ δ, so that for every α ∈ Lim(C) ∩ Z, C α = C ∩ α. The statement that there is a coherent sequence on δ that cannot be threaded is denoted P(δ).  ...  Then there is a forcing extension which satisfies the following: for all n < ω, Col(δ,δ +(n) ) "δ is weakly compact and that for every set Z in the Woodin filter for δ the weak compactness of δ can be  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00153-015-0465-4">doi:10.1007/s00153-015-0465-4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nlk4dcs2qjfuhg6hmk4yhuxjya">fatcat:nlk4dcs2qjfuhg6hmk4yhuxjya</a> </span>
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A model of the generic Vopěnka principle in which the ordinals are not Mahlo [article]

Victoria Gitman, Joel David Hamkins
<span title="2018-04-15">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo.  ...  Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a Δ_2-definable class containing no regular cardinals.  ...  , Gitman and Schindler proved that the generic form gVP(Σ n+2 ) is equiconsistent with a proper class of virtually C (n) -extendible cardinals.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1706.00843v2">arXiv:1706.00843v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2cmbd5egpzhstmafjxfe7lvqli">fatcat:2cmbd5egpzhstmafjxfe7lvqli</a> </span>
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Model Theoretic Characterizations of Large Cardinals Revisited [article]

Will Boney, Stamatis Dimopoulos, Victoria Gitman, Menachem Magidor
<span title="2022-02-01">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We continue this work, by establishing such characterizations for Woodin cardinals (and variants), various virtual large cardinals, and subtle cardinals.  ...  In [Bon20], model theoretic characterizations of several established large cardinal notions were given.  ...  For example, ω is the strong compactness cardinal of first-order logic, a weakly compact cardinal κ is a weak compactness cardinal of the infinitary logic L κ,κ , and a strongly compact cardinal κ is a  ... 
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