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Generalizing the Mahlo Hierarchy, with Applications to the Mitchell Models

Stewart Baldwin
1983 Annals of Pure and Applied Logic  
The process can then be repeated to the next stage to (possibly) get a higher order Mahlo cardinal K such that {a < K: a is weakly-Mahlo} is stationary, and so forth.  ...  We assume the reader is familiar with the terms closed, unbounded, and stationary for subsets of a regular cardinal.  ...  The generalized Mahlo hierarchy When it comes to carrying the ideas of Definition 2.15 further, there is more than one way to proceed.  ... 
doi:10.1016/0168-0072(83)90010-6 fatcat:a24hs3zvwjeu3d5w7mxa47w6ai

Force to change large cardinal strength [article]

Erin Carmody
2015 arXiv   pre-print
the hyper-inaccessible hierarchy.  ...  I also show that a cardinal κ which is Mahlo in the ground model can have every possible inaccessible degree in the forcing extension, but no longer be Mahlo there.  ...  One might be tempted to define the next degree of Mahlo to be a Mahlo limit of Mahlo cardinals, exactly as was done with the degrees of inaccessible cardinals, and indeed there is a hierarchy of Mahlo  ... 
arXiv:1506.03432v2 fatcat:67wxk57rabgwpbolcuoxsxslj4

Page 4638 of Mathematical Reviews Vol. , Issue 2002G [page]

2002 Mathematical Reviews  
Woodin constructed a model in which ©, fails for a Mahlo cardinal «, by forcing over a model with a hypermeasurable / of Mitchell order slightly more than o(A) = A**.  ...  They also establish a stronger result, generalizing from 2"° to general x.  ... 

Page 1318 of Mathematical Reviews Vol. , Issue 85d [page]

1985 Mathematical Reviews  
Baldwin, Stewart (1-ABRN) 85d:03106 Generalizing the Mahlo hierarchy, with applications to the Mitchell models. Ann. Pure Appl. Logic 25 (1983), no. 2, 103-127.  ...  The author generalizes Mahlo’s hierarchy and uses these new Mahlo operations to define canonical inner models having many measurable cardinals.  ... 

Announcement

D. van Dalen
1986 Annals of Pure and Applied Logic  
20 (1981) 289-330 Baldwin, S., Generalizing the Mahlo hierarchy, with applications to the Mitchell models 25 (1983) 103-127 Barwise, J., M.  ...  Shelah, Narrow Boolean algebras 28 (1985) 1-12 Bruce, K.B., Model-theoretic forcing in logic with a generalized quantifier 13 (1978) 225-265 Buechler, S., Locally modular theories of finite rank  ... 
doi:10.1016/0168-0072(86)90027-8 fatcat:r7qblmmvnzdvjfg534p2d7jfqi

JSL volume 39 issue 1 Cover and Front matter and Errata

1974 Journal of Symbolic Logic (JSL)  
MITCHELL 57 A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property. By D. M. GABBAY and D. H. J. D E JONGH 67 PCA well-orderings of the line.  ...  MITCHELL A sequence of decidable finitely axiomatizable intermediate logics with thedisjunction property. By D. M. GABBAY and D. H. J. D E JONGH PCA well-orderings of the line.  ...  Linear sampling and the V3V case of the decision problem. By STAL O. AANDERAA  ... 
doi:10.1017/s0022481200065336 fatcat:okv4hkahrvf5vc6tuzsrqltbau

JSL volume 39 issue 2 Cover and Front matter and Errata

1974 Journal of Symbolic Logic (JSL)  
MITCHELL 57 A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property. By D. M. GABBAY and D. H. J. D E JONGH 67 PCA well-orderings of the line.  ...  MITCHELL A sequence of decidable finitely axiomatizable intermediate logics with thedisjunction property. By D. M. GABBAY and D. H. J. D E JONGH PCA well-orderings of the line.  ...  Linear sampling and the V3V case of the decision problem. By STAL O. AANDERAA  ... 
doi:10.1017/s0022481200064343 fatcat:5gexvldlkzbufc2ygfii7gdrgy

Page 2244 of Mathematical Reviews Vol. , Issue 88e [page]

1988 Mathematical Reviews  
This casts further light on the known con- nections to the Mahlo hierarchy and the Mitchell order of normal ulrafilters.  ...  (H-AOS) A remark on partition relations for infinite ordinals with an application to finite combinatorics. Logic and combinatorics (Arcata, Calif., 1985), 157-167, Contemp. Math., 65, Amer. Math.  ... 

JSL volume 66 issue 4 Cover and Back matter

2001 Journal of Symbolic Logic (JSL)  
Number systems with simplicity hierarchies: a generalization of Conway's theory of surreal numbers 1231 ENAYAT, ALL Power-like models of set theory 1766 The consistency strength of successive cardinals  ...  Expansions of dense linear orders with the intermediate value property 1783 MILLER, RUSSELL. The A"-spectrum of a linear order 470 MITCHELL, W. P. R. and SIMMONS, H.  ...  Requests for information, applications for membership, orders for back volumes, business correspondence, and notices and announcements for publication in the BULLETIN should be sent to the Secretary-Treasurer  ... 
doi:10.1017/s0022481200010124 fatcat:dvgtmcubczd6nm5nvjjz6tvjnq

Large Cardinals with Forcing [chapter]

Akihiro Kanamori
2012 Handbook of the History of Logic  
Through this work the concept of saturated ideal, first studied by Tarski, was brought to prominence as a generalization of having a measurable cardinal applicable to the low levels of the cumulative hierarchy  ...  Mitchell also observed that if there are no special ω 2 -Aronszajn trees, then ω 2 is Mahlo in L.  ... 
doi:10.1016/b978-0-444-51621-3.50004-9 fatcat:rxofgx3z6nelvlyhmdybzlox6e

Large cardinals and their effect on the continuum function on regular cardinals

Radek Honzík
2016 Acta Universitatis Carolinae: Philosophica et Historica  
The paper concludes with a result from the opposite end of the spectrum: namely, how to kill all large cardinals in the universe.  ...  In particular, we will consider inaccessible, Mahlo, weakly compact, Ramsey, measurable, strong, Woodin, and supercompact cardinals.  ...  Acknowledgments The author was supported by GAČR project I 1921-N25.  ... 
doi:10.14712/24647055.2016.9 fatcat:v6jtg6naezhwrpi5cobyubbuei

Coding over Core Models [article]

Sy David Friedman and Ralf Schindler and David Schrittesser
2022 arXiv   pre-print
In this article we aim to survey some of the work that has been done which combines these two themes, extending Jensen's original Coding Theorem from L to core models witnessing large cardinal properties  ...  Early in their careers, both Peter Koepke and Philip Welch made major contributions to two important areas of set theory, core model theory and coding, respectively.  ...  Measures of higher order Suppose now that we are in a "Mitchell model" L[E] where we now have two normal measures U 0 , U 1 on κ with U 0 below U 1 in the Mitchell order.  ... 
arXiv:2209.08696v1 fatcat:vw6j4r6c2zcydatwepxzsvlvjy

Global singularization and the failure of SCH

Radek Honzik
2010 Annals of Pure and Applied Logic  
Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then there exists a cardinal-preserving  ...  Great thanks are due to my advisor Sy D. Friedman who gave me guidance and a lot of help. I also wish to thank James Cummings, Moti Gitik and Menachem Magidor for the personal discussions.  ...  The work was supported by a postdoctoral grant of the Grant Agency of the Czech Republic 201/09/P115, and by a Short Visit Grant 2903 from the European Science Foundation: New frontiers of infinitymathematical  ... 
doi:10.1016/j.apal.2009.11.002 fatcat:epbfoytlhrcsrfsvxlfeu7sanm

Inner Models for Large Cardinals [chapter]

William J. Mitchell
2012 Handbook of the History of Logic  
Mitchell Constructibility According to Kanamori [Kanamori, 2010b], "An inner model is a transitive class containing all the ordinals such that, with membership and quantification re-3 stricted to it, the  ...  The model HOD has 12 no intrinsic structure: the cumulative hierarchy in V is needed to to see that the class HOD is even definable, and moreover the formula defining it is not absolute, so that the model  ...  We begin with an innovation which, though inessential and technical, was important to the later study of L and of fine structure models in general: the introduction of the J α hierarchy in place of the  ... 
doi:10.1016/b978-0-444-51621-3.50005-0 fatcat:m3yjj6pueveizmlqy4dkfp5lju

Characterizing large cardinals through Neeman's pure side condition forcing

Peter Holy, Philipp Lücke, Ana Njegomir
2020 Fundamenta Mathematicae  
Our characterizations make use of the concepts of internal large cardinals introduced in this paper, and of the classical concept of generic elementary embeddings. 2020 Mathematics Subject Classification  ...  The combinatorial properties that we make use of are natural principles, and in particular for inaccessible cardinals, these principles are equivalent to their corresponding large cardinal properties.  ...  If θ is not a Mahlo cardinal, then there is an inner model that contains a stationary limit of measurable cardinals of uncountable Mitchell order. Proof.  ... 
doi:10.4064/fm662-1-2020 fatcat:lapwdwoyhzggpkbg6odqdg55wu
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