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Gunter Fuchs The College of Staten Island (CUNY) Department of Mathematics 2800 Victory Boulevard Staten Island, NY 10314 room 1S-225 phone: 718-982-4182 math department: 718-982-3600 The Graduate Center (CUNY) Mathematics Program 365 5th Avenue New York, NY 10016 room 4432 phone: 212-817-8143 math program: 212-817-8530 Links: Researchgate Google Scholar NYLogic website NYLogic YouTube channel

Please send any questions or comments to:

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Teaching

In the Fall 2024 semester, I teach MTH 229 and MTH 231 at CSI, and a topics course on Forcing in Set Theory at the CUNY Graduate Center.

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Research Interests

• Set Theory
• Inner Model Theory
• Forcing, Forcing Axioms, Iterated Forcing

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Publications and Preprints

Please note that most of the preprints that are available here differ from the published versions.

Published

• Gunter Fuchs. On the consistency strength of critical leaps.
  Archive for Mathematical Logic (2024, online). Open access view only version here. DOI: 10.1007/s00153-024-00951-4
  
• Gunter Fuchs. Errata: on the role of the continuum hypothesis in forcing principles for subcomplete forcing.
  Archive for Mathematical Logic 63: 509-521 (2024). DOI: 10.1007/s00153-024-00905-w
  
• Sean Cox and Gunter Fuchs. The diagonal strong reflection principle and its fragments.
  Journal of Symbolic Logic 88(3): 1281-1309 (2023). DOI: 10.1017/jsl.2023.3
  Preprint on the arXiv.
• Gunter Fuchs. Blurry definability.
  Mathematics 10 (2022). Special issue "Mathematical Logic and Its Applications 2021". Open access.
  DOI: 10.3390/math10030452
• Gunter Fuchs. Canonical fragments of the strong reflection principle.
  Journal of Mathematical Logic 21(3): 2150023 (2021). 58 pages.
  Preprint on the arXiv.
• Gunter Fuchs. Aronszajn tree preservation and bounded forcing axioms.
  Journal of Symbolic Logic 86(1): 293-315 (2021). DOI: 10.1017/jsl.2021.13
  Preprint on the arXiv.
• Gunter Fuchs and Chris Lambie-Hanson. Separating diagonal stationary reflection principles.
  Journal of Symbolic Logic 86(1): 262-292 (2021). DOI: 10.1017/jsl.2020.56
  Preprint on the arXiv.
• Arthur Apter, Shoshana Friedman and Gunter Fuchs. More on HOD-supercompactness.
  Annals of Pure and Applied Logic 172(3): 21 pages (2021). DOI: 10.1016/j.apal.2020.102901
  Preprint.
• Neil Barton, Andrés Eduardo Caicedo, Gunter Fuchs, Joel David Hamkins, Jonas Reitz and Ralf Schindler. Inner-model reflection principles.
  Studia Logica 108: 573-595 (2020). DOI: 10.1007/s11225-019-09860-7
  Preprint available at the arXiv.
• Gunter Fuchs. Diagonal reflections on squares.
  Archive for Mathematical Logic 58(1-2): 1-26 (2019). DOI: 10.1007/s00153-018-0614-7
  Here is an open access version (read only), and here is a preprint.
• Gunter Fuchs. Subcomplete forcing principles and definable well-orders.
  Mathematical Logic Quarterly 64(6):487-504 (2018). DOI: 10.1002/malq.201800008
  Preprint: arXiv:1708.08167 [math.LO]
• Gunter Fuchs and Ralf Schindler. The solidity and nonsolidity of initial segments of the core model.
  Journal of Symbolic Logic 83(3): 920-938 (2018). DOI: 10.1017/jsl.2018.45
  Here is a preprint.
• Gunter Fuchs and Kaethe Minden. Subcomplete forcing, trees and generic absoluteness.
  Journal of Symbolic Logic 83(3): 1282-1305 (2018). DOI: 10.1017/jsl.2018.23
  Preprint here and on the arXiv:1708.08170 [math.LO]
• Gunter Fuchs. Closure properties of parametric subcompleteness.
  Archive for Mathematical Logic 57(7-8): 829-852 (2018). DOI: 10.1007/s00153-018-0611-x
  Here is the online first version (read only), and here is a preprint.
• Gunter Fuchs and Assaf Rinot. Weak square and stationary reflection.
  Acta Mathematica Hungarica 155(2): 393-405 (2018). DOI: 10.1007/s10474-018-0789-8.
  Here is the online first version (read only), and here is a preprint. Also available on the arxiv.
• Gunter Fuchs, Victoria Gitman and Joel David Hamkins. Ehrenfeucht's lemma in set theory.
  Notre Dame Journal of Formal Logic 59(3): 355-370 (2018). DOI: 10.1215/00294527-2018-0007.
  Preprint available at the arXiv.
• Gunter Fuchs. Hierarchies of forcing axioms, the continuum hypothesis and square principles.
  Journal of Symbolic Logic 83(1): 256-282 (2018). DOI: 10.1017/jsl.2017.46
  Here is a preprint.
• Gunter Fuchs. Hierarchies of (virtual) resurrection axioms.
  Journal of Symbolic Logic 83(1): 283-325 (2018). DOI: 10.1017/jsl.2017.65
  Here is a preprint (updated on 8/23/2017).
• Gunter Fuchs. The subcompleteness of Magidor forcing.
  Archive for Mathematical Logic 57(3-4): 273-284 (2018). DOI: 10.1007/s00153-017-0568-1
  Here is a read only version online (free access), and here is a preprint.
• Gunter Fuchs, Victoria Gitman and Joel David Hamkins. Incomparable ω₁-like models of set theory.
  Mathematical Logic Quarterly 63 (1-2):66-76 (2017). DOI: 10.1002/malq.201500002
  Preprint available at the arXiv.
• Gunter Fuchs and Ralf Schindler. Inner model theoretic geology.
  Journal of Symbolic Logic 81(3):972-996 (2016). DOI: 10.1017/jsl.2015.64
  Here is a preprint.
• Gunter Fuchs. The stationarity of the collection of the locally regulars.
  Archive for Mathematical Logic 54(5):725-739 (2015). DOI: 10.1007/s00153-015-0437-8
  Here is a preprint.
• Gunter Fuchs, Joel David Hamkins and Jonas Reitz. Set-theoretic geology.
  Annals of Pure and Applied Logic. 166(4):464-501 (2015). DOI: 10.1016/j.apal.2014.11.004.
  Preprint available on the arXiv.
• Gunter Fuchs. On sequences generic in the sense of Magidor.
  Journal of Symbolic Logic 79(4):1286-1314 (2014). DOI: 10.1017/jsl.2014.33.
  Here is a preprint.
• Gunter Fuchs. Club degrees of rigidity and almost Kurepa trees.
  Archive for Mathematical Logic 52(1): 47-66 (2013). DOI: 10.1007/s00153-012-0306-7.
  Here is a preprint.
• Gunter Fuchs and Philipp Lücke. Iteratively changing the heights of automorphism towers.
  Notre Dame Journal of Formal Logic 53(2):155-174 (2012). DOI: 10.1215/00294527-1715662.
  Here is a preprint.
• Gunter Fuchs. λ-structures and s-structures: translating the iteration strategies.
  Annals of Pure and Applied Logic 162(9):710-751 (2011). DOI: 10.1016/j.apal.2011.02.002.
  Here is a preprint.
• Gunter Fuchs. λ-structures and s-structures: translating the models.
  Annals of Pure and Applied Logic 162(4):257-317 (2011). DOI: 10.1016/j.apal.2010.08.003.
  Here is a preprint.
• Gunter Fuchs. Generic embeddings associated to an indestructibly weakly compact cardinal.
  Annals of Pure and Applied Logic 162(1):89-105 (2010). DOI: 10.1016/j.apal.2010.07.005.
  Here is a preprint.
• Gunter Fuchs, Itay Neeman, and Ralf Schindler. A criterion for coarse iterability.
  Archive for Mathematical Logic 49(4):447-467 (2010).
  This article is also available electronically on SpringerLink (open access, DOI: 10.1007/s00153-010-0180-0).
• Gunter Fuchs. Combined maximality principles up to large cardinals.
  Journal of Symbolic Logic 74(3):1015-1046 (2009).
  Here is a preprint.
• Gunter Fuchs and Joel David Hamkins. Degrees of rigidity for Souslin trees.
  Journal of Symbolic Logic 74(2):423-454 (2009).
  Preprint available at the arXiv.
• Gunter Fuchs. Successor levels of the Jensen Hierarchy.
  Mathematical Logic Quarterly 55(1):53-69 (2009).
  Here is a preprint.
• Gunter Fuchs and Joel David Hamkins. Changing the heights of automorphism towers by forcing with Souslin trees over L.
  Journal of Symbolic Logic 73(2):614-633 (2008).
  Preprint available at the arXiv.
• Gunter Fuchs. Closed maximality principles: implications, separations and combinations.
  Journal of Symbolic Logic 73(1):276-308 (2008).
  Here is a preprint.
• Gunter Fuchs. A characterization of generalized Příkrý sequences.
  Archive for Mathematical Logic 44(8):935-971 (2005).

Submitted

• Gunter Fuchs and Kaethe Minden. Split Principles.
  To appear in the Notre Dame Journal of Formal Logic. Preprint on the arXiv.
• Gunter Fuchs and Corey Bacal Switzer. Iteration theorems for subversions of forcing classes.
  To appear in the Journal of Symbolic Logic. Submitted in 2020. 45 pages.
  Preprint on the arXiv.
• Gunter Fuchs. More on blurry HOD.
  Submitted in 2023. 32 pages.
  

Other works

• Gunter Fuchs. λ-Strukturen und s-Strukturen.
  Dissertation (in German), Humboldt University Berlin. DOI: 10.18452/14903. 297 pages (2003).
• Gunter Fuchs. Eine Verallgemeinerung von Prikrys Forcing.
  Master's Thesis (German Diplom, written in German), Humboldt University Berlin. Here is a copy. 78 pages (1998).

In preparation

• Gunter Fuchs and Kaethe Minden. Split principles, large cardinals, splitting families and split ideals.
  Preprint on the arXiv.
• Gunter Fuchs and Joel David Hamkins. Boolean ultrapowers, the Bukovský-Dehornoy phenomenon, and iterated ultrapowers.
  Preprint on the arxiv.
• Gunter Fuchs. The strong Příkrý property.
  Preprint on the arxiv.

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Selected Talks

• Blurry HOD. A hierarchy of inner models.
  Rutgers logic seminar.
  February 12, 2024.
• On the structure of the blurry HOD hierarchy.
  Kobe-Waseda joint set theory seminar.
  November 2, 2023.
• On a fragment of the strong reflection principle consistent with CH.
  RIMS Set theory workshop, Kyoto University, Japan.
  October 24, 2023.
• On the strength of leap constellations.
  Invited talk at From ω to Ω, National Unversity of Singapore.
  July 5, 2023.
• Blurry HOD. A sketch of a landscape.
  Plenary talk at the Logic Colloquium 2022, Reykjavík University, Iceland.
  June 29, 2022.
• Blurry definability.
  Virtual talk at the set theory seminar of the Kurt Gödel Research Center, Vienna.
  January 18, 2022.
• Spread out sets and the subcomplete fragment of SRP
  Virtual talk held at Gemeinsame Jahrestagung der DMV und der ÖMG.
  September 28, 2021.
• Fragments of (diagonal) strong reflection
  Two part talk held at the virtual Münster Set Theory Research Seminar.
  June 2 and 9, 2021.
• The subcomplete fragment of the Strong Reflection Principle
  Virtual Carnegie Mellon University Mathematical Logic Seminar.
  September 29, 2020.
• Canonical fragments of the Strong Reflection Principle
  Virtual CUNY Set Theory Seminar.
  August 14, 2020.
  This talk was recorded and can be watched on the NYLogic YouTube channel.
• Relativizing strong reflection principles to forcing classes
  15th international Luminy workshop in set theory, CIRM Luminy, France.
  September 25, 2019.
  
• Simultaneous exact reflection and mutual stationarity
  CUNY Set Theory Seminar, CUNY Graduate Center, New York.
  September 13, 2019.
  
• Ordinal diagonal reflection principles and squares
  Seventh European Set Theory Conference, University of Vienna, Austria.
  July 1, 2019.
  
• Bounded forcing axioms and the preservation of wide Aronszajn trees
  CUNY Set Theory Seminar, CUNY Graduate Center, New York.
  February 1, 2019.
  
• Definable well-orderings of the reals from subcomplete forcing principles
  Set Theory Seminar, University of Bonn, Germany.
  January 15, 2019.
  
• What is the bounded forcing axiom for countably closed forcing?
  CUNY Set Theory Seminar, CUNY Graduate Center, New York.
  August 31, 2018.
  
• Subversions of iterable forcing classes
  First Girona conference on inner model theory, Girona (Catalonia).
  July 19, 2018.
  
• Determining the extent of weak Todorcevic square principles by reflecting diagonally
  CUNY Set Theory Seminar, CUNY Graduate Center, New York.
  November 17, 2017.
  
• Subcomplete forcing and its forcing principles
  Set theory conference on the occasion of Ronald Jensen's 80th birthday, Münster, Germany.
  August 3rd, 2017.
  Slides and an overview diagram
• Consequences of subcomplete forcing principles
  CUNY Logic Workshop, CUNY Graduate Center, New York.
  April 21, 2017.
• Weak bounded forcing axioms and remarkably reflecting cardinals
  CUNY Set Theory Seminar, CUNY Graduate Center, New York.
  March 3, 2017.
• Hierarchies of forcing axioms
  CUNY Logic Workshop, CUNY Graduate Center, New York.
  November 4, 2016.
• Boolean Ultrapowers in Set Theory
  CUNY Logic Workshop, CUNY Graduate Center, New York.
  February 5, 2016.
• Boolean ultrapowers and the Bukovsky-Dehornoy phenomenon
  CUNY Set Theory Seminar, CUNY Graduate Center, New York.
  September 18, 2015.
• Prikry-type sequences: a composition of interconnected results
  The fifth European Set Theory Conference, Cambridge, UK.
  August 27, 2015. Here is a video of the talk.
• Set-theoretic geology in the presence of an inner model with a Woodin cardinal
  3rd Münster conference on inner model theory, the core model induction and hod mice, WWU Münster, Germany.
  July 20, 2015.
• News on the Solid Core
  CUNY Set Theory Seminar, CUNY Graduate Center, New York.
  November 7, 2014.
• Algebraicity versus Definability
  CUNY Set Theory Seminar, CUNY Graduate Center, New York.
  April 25, 2014.
• Prikry-type sequences, iterations and critical sequences
  CUNY Logic Workshop, CUNY Graduate Center, New York.
  November 8, 2013.
• Characterizing Magidor Sequences
  Rutgers Logic Seminar, Rutgers University, New Jersey.
  September 16, 2013.
• Magidor Sequences
  Set Theory Seminar Seminar, Münster University, Germany.
  July 9, 2013.
• Constructing a Souslin almost Kurepa tree
  CUNY Set Theory Seminar, CUNY Graduate Center, New York.
  March 23, 2012.
• The Stationarity of the Collection of Local Aleph Ones
  CUNY Set Theory Seminar, CUNY Graduate Center, New York.
  February 24, 2012.
• On Changing Automorphism Tower Heights by Forcing
  Rutgers Logic Seminar, Rutgers University, New Jersey.
  February 20, 2012.
• Set-theoretic Geology and Canonical Inner Models
  CUNY Logic Workshop, CUNY Graduate Center.
  September 23, 2011.
• Mysteries of the Generic Multiverse
  Toronto Set Theory Seminar, Fields Institute, Toronto.
  August 19, 2011.
• Automorphisms of Souslin Trees
  Summer Conference on General Topology and its Applications, The City College of New York.
  July 27, 2011.
• An Application of Almost Disjoint Coding
  CUNY Set Theory Seminar, CUNY Graduate Center, New York.
  May 6, 2011.
• Structural Consequences of AD I+II
  CUNY Set Theory Seminar, CUNY Graduate Center, New York.
  September 3 and September 14, 2010.
• Iterated Ultrapowers and Inner Models of Measurability I+II
  CUNY Set Theory Seminar, CUNY Graduate Center, New York.
  April 23 and April 30, 2010.
• Reflections on the Structure of the Generic Multiverse
  Rutgers Logic Seminar, Rutgers University, New Jersey.
  March 29, 2010.
• Some recent forcing ideas
  Lecture series held at Universidad Autónoma Metropolitana, Unidad Iztapalapa in Mexico City.
  January 11 to January 15, 2010.
• An Invitation to Inner Model Theory
  CUNY Set Theory Seminar, CUNY Graduate Center, New York.
  November 6 to November 20, 2009.
• Set Theoretic Geology meets Inner Model Theory
  Mini-workshop on Fine Structure Theory, Oberwolfach, Germany.
  November 10, 2009.
• Maximality Principles and the Generic Multiverse
  CUNY Logic Workshop, CUNY Graduate Center, New York.
  September 25, 2009.
• The Solovay Model
  Talk at my Habilitation Colloquium, Münster University. This talk was aimed at general mathematicians.
  July 22, 2009. By giving this talk, I completed my Habilitation.
• Fixpunkte: Eine fixe Idee in der Mathematik
  Talk at the Lange Nacht der Mathematik, Münster University. This talk was aimed at a general audience of non-mathematicians.
  May 30, 2008. Due to public request, the slides for this talk can be downloaded in pdf format.
• Closed Maximality Principles, Large Cardinals, etc.
  CUNY Set Theory Seminar, The CUNY Graduate Center, New York.
  February 29, 2008.
• From Maximality Principles Via Weak Compactness to Generic Embeddings
  Young Set Theory workshop in Bonn, Germany,
  January 24, 2008.
• Aspects of Indestructible Weak Compactness
  Set Theory Workshop 2008 at the Mathematical Research Institute Oberwolfach, Germany,
  January 15, 2008.
• Maximality Principles for Closed Forcings
  Logic Colloquium 2007 in Wroclaw, Poland
  July 18, 2007. Due to public request, this talk can be downloaded in pdf format.
• Closed Maximality Principles and Large Cardinals
  First European Set Theory Meeting in Bedlewo, Poland
  July 12, 2007. This talk can also be downloaded in pdf format.
• Maximality Principles for Closed Forcings
  Lecture at the Set Theory Seminar in Bonn, Germany
  April 10, 2007.
• Degrees of Rigidity and Changing the Heights of Automorphism Towers
  Colloquium Logicum 2006/Annual Meeting of the German Mathematicians' Association in Bonn, Germany
  September 22, 2006
• Degrees of Rigidity for Souslin Trees
  Mini-Workshop on Fine Structure and Inner Models at the Mathematical Research Institute Oberwolfach, Germany
  May 1, 2006.
• λ-Structures and s-Structures
  Lecture at the Computational Prospects of Infinity program at the National University of Singapore
  July 8, 2005.
• Degrees of Rigidity for Souslin Trees
  Lecture at the Computational Prospects of Infinity program at the National University of Singapore
  June 24, 2005.
• λ-Strukturen und s-Strukturen
  Berlin-Potsdam Logic Colloquium, Berlin, Germany
  December 9, 2002
• Mitchell-Steel Mice versus Friedman-Jensen Mice
  Mini-Workshop on Fine Structure Theory at the Mathematical Research Institute Oberwolfach, Germany
  February 26, 2001
• Characterizing Generalized Příkrý Sequences
  Lecture at the Set Theory Seminar, Bonn University, Germany
  June 29, 2000
• A Generalization of Příkrý Forcing
  Berlin-Potsdam Logic Colloquium, Berlin, Germany
  January 10, 2000.

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Vitae

• Born in Västanfors, Sweden
• Starting August 2019: Full Professor at CSI / CUNY
• Starting August 2013: Associate Professor at CSI / CUNY
• December 2011: Joined the doctoral faculty at the Ph.D. Program in Mathematics of the CUNY Graduate Center
• Starting September 2009: Assistant Professor at CSI / CUNY
• July 22, 2009: Habilitation
• Dec. 2004 - August 2009: Assistant Professor at Münster University (German wissenschaftlicher Assistent), Germany
• June 2003: Received Ph. D. in Mathematics, supervisor: Ronald Jensen (Humboldt University Berlin)
• Oct. 1998 - November 2004: DFG fellow in Berlin/Münster
• Sept. 1998: Received Diploma (M. Sc.) in Mathematics (Humboldt University Berlin)
• Oct. 1995 - Sept. 1998: Graduate Assistant (Humboldt University Berlin)