Author Archives: Jonas Reitz

My Mathematical Geneology

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Inner-Model Reflection Principles

Abstract. We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \phi(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W\subsetneq V. A stronger principle, the ground-model reflection principle, asserts that any such \phi(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy-Montague reflection theorem. They are each equiconsistent with ZFC and indeed \Pi_2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles.


Inner-model reflection principles, with Neil Barton, Andres Caicedo, Gunter Fuchs, Joel David Hamkins and Ralf Schindler, Stud Logica (2019).  pdf, arXiv


Subversive Compromise

Subversive Compromise:
Working in and around the system to write a grant, get funded, and make stuff happen.

Metro NExT (New Experiences in Teaching) Workshop 2019
September 13, 2019 – Courant/NYU (251 Mercer St.), Room 109

In this interactive workshop, participants will explore challenges, opportunities and lessons learned in conceiving, planning, designing, and writing a successful grant proposal.


Generalized Cohen Iterations

Set Theory Seminar (RESCHEDULED from April 12)
CUNY Graduate Center, Room 6417
Friday, May 17, 10:00-11:45am
Jonas Reitz, CUNY
Generalized Cohen Iterations

Adding Cohen subsets to each of a class of cardinals in turn is a common construction in set theory, and underlies many fundamental results. The construction comes in two basic flavors, products (as in Easton’s Theorem on the powers of regular cardinals) and iterations (forcing the GCH). These flavors are apparently quite similar, forcing at stage kappa to add subsets via the Cohen partial order Add(kappa,lambda). They differ only in the universe over which Add(kappa,lambda) is defined – in the case of products the ground model poset is used at each stage, whereas in typical iterations the poset is taken from the partial extension up to kappa. In this talk I will consider an alternative, in which we allow Add(kappa,lambda) to be defined over an arbitrary inner model (lying between the ground model and the extension up to kappa) at each stage. These generalized Cohen iterations are ZFC-preserving, although neither the proof for products nor for traditional iterations transfers directly. They allow constructions such as class iterations of class products of Cohen forcing, with applications including new work with Kameryn Williams on iterating the Mantle.


Opening Gateways: Supporting Success in STEM through Success in Gateway Mathematics

CUNY CUE Conference 2019
May 10, 2019
New York City College of Technology’s new Academic Complex, 285 Jay Street

Presenters:  Jonas Reitz, Charlie Edwards, Laura Ghezzi, Andrew Parker

Abstract: “Opening Gateways to Completion: Open Digital Pedagogies for Student Success in STEM” is a 5 year collaborative grant between the New York City College of Technology and the Borough of Manhattan Community College funded through the Department of Education’s Developing Hispanic-Serving Institutions (Title V) program.  The project supports student success in mathematics courses that serve as gateways to STEM disciplines, courses that often act as barriers to progress and completion in these disciplines. Cohorts of full-time and part-time faculty from both campuses take part in an intensive professional development seminar, where they are exposed to active learning strategies, open digital pedagogies, multiplayer and flipped classroom techniques, games in the classroom, WeBWorK, Desmos and much more.

An ecosystem of high-quality OERs support the pedagogy of our participants. WeBWorK, an open source alternative to expensive and proprietary online homework systems, serves as a platform for the development of problems and problem sets aligned with the curriculum, with customized feedback and error-recognition.  At City Tech, additional development has bridged WeBWorK and the OpenLab: students seeking help on WeBWorK are directed to an OpenLab community space where they can ask and answer questions. Classroom activities and STEM applications developed by our participants, and a curated collection of resources such as online videos, round out our OER ecosystem.

Join us to learn more about our professional development model and see first hand some of the exciting OERs, activities and STEM applications developed and utilized by our participants for Algebra and Precalculus courses.

WeBWorK on the OpenLab:  Leveraging City Tech’s open digital platform to create a community space for homework help

Venue: Mathematics Association of America (MAA)
Metro New York Section, 2019 Annual Meeting
New York City College of Technology, May 4, 2019

Presenters:  Andrew Parker, Charlie Edwards, Jonas Reitz

Abstract: WeBWorK, an open-source online homework system supported by the MAA and the NSF, provides a platform for students to practice and engage with their mathematics studies.   WeBWorK offers a number of advantages over traditional pencil-and-paper homework, including instant, customized feedback and error-recognition. But how do we help students when they get stuck? At City Tech a team of faculty and developers has worked to bridge WeBWorK and the OpenLab, our open digital platform for teaching, learning and collaboration.  Students seeking help on a WeBWorK problem are directed to an OpenLab community space where they can review answers to previous questions about their problem or ask their own. By moving the conversations around homework help into a public space we increase transparency, reducing the repetitious explanations that can occur in one-on-one support models, such as email, where many students can ask very similar questions, each requiring a near-identical response from the instructor.

Want to bring this technology to your own institution? We will discuss how you can set up your own OpenLab, free, through Commons in a Box OpenLab, and our planned release of the WeBWorK OpenLab bridge.

Join us to learn more and see the project in action.


Opening Gateways through WeBWorK: Expanding the power of online homework to build engagement and support student learning

Venue: Mathematics Association of America (MAA)
Metro New York Section, 2019 Annual Meeting
New York City College of Technology, May 4, 2019

Presenters:  Andrew Parker, Marianna Bonanome, Ariane Masuda, Jonas Reitz

Abstract:  WeBWorK provides a powerful platform for online homework, freely available and open source.  At City Tech, more than a decade of experience with WeBWorK has provided many opportunities to experiment with existing features, explore new functionality, and develop new tools that extend the reach and scope of the WeBWorK system.  

In addition to developing high-quality problems and problem sets aligned with the curriculum, our work makes frequent use of customized feedback and error-recognition, providing guidance without giving away too much about the expected result.  Cutting edge features such as scaffolded problems provide ongoing support to students as they progress through multi-step solutions, and Just-In-Time problem sets adapt to student behavior, providing practice at the point of greatest need. Integration of GeoGebra, another open source mathematics platform, into WeBWorK problems provides a rich visual dimension to mathematical problem-solving.  Finally, the development of our WeBWorK leaderboards project brings the motivating principles of game-based learning to the WeBWorK platform, with exciting preliminary results.

Join us in this session as we share a multitude of examples as well as lessons learned and best practices for WeBWorK development.

Gödel’s Incompleteness – the most abused theorem in modern mathematics

New York City College of Technology Mathematics Seminar
Thursday, April 11, 2019, 12:45pm-2pm, Room N700

“It is sometimes claimed to prove the existence of God or of free will, the necessary incompleteness of the Bible or of the U.S. Constitution, or the impossibility of genuine knowledge in mathematics—just to mention a few of the many alleged applications”

– Panu Raatikainen, Notices of the AMS, March 2007

The goal of this talk is to give an overview of Gödel’s First Incompleteness Theorem, exploring the precise formulation of this famous result.  The talk will include an introduction to Mathematical Logic and touch on some of the historical events in the field surrounding Gödel’s work.

Slides for the talk can be found here.


Cohen forcing and inner models, (2020). Math. Log. Quart., 66: 65-72. doi:10.1002/malq.201800062

Inner mantles and iterated HOD, with Kameryn J. Williams, (2019), . Math. Log. Quart., 65: 498-510. doi:10.1002/malq.201800071.  arXiv

Inner-model reflection principles, with Neil Barton, Andres Caicedo, Gunter Fuchs, Joel David Hamkins and Ralf Schindler, Stud Logica (2019).  pdf, arXiv

From Geometry to Geology: An Invitation to Mathematical Pluralism Through the Phenomenon of Independence, Journal of Indian Council of Philosophical Research, 34(2), 2017, 289-308. doi:10.1007/s40961-017-0099-5.  pdf, arXiv

Set-theoretic geology, with Gunter Fuchs and Joel David Hamkins, The Annals of Pure and Applied Logic, 166(4), April 2015, 464-501. arXiv

Pointwise Definable Models of Set Theory, with Joel David Hamkins and David Linetsky, The Journal of Symbolic Logic, 78(1), March 2013, 139-156

The Ground Axiom is Consistent with V \neq HOD, with Joel David Hamkins and W. Hugh Woodin, Proceedings of the American Mathematical Society, 136(8), August 2008, 2943-2949.

The Ground Axiom, The Journal of Symbolic Logic, 72(4), 2007, 1299-1317.



My name is Jonas Reitz, and I am a Professor in the Mathematics Department at City Tech.

My research interests are in mathematical logic, in particular set theory, in more particular the method of universe-construction developed by Paul Cohen known as forcing.

I teach many different courses, although in recent years I’ve focused on three: MAT 2571 Introduction to Proofs (formerly 2071), MAT 2680 Differential Equations, and MAT 1375 Precalculus.

Over the past near-decade, much of my institutional work at City Tech has revolved around two major Title V grant projects, the Living Lab and Opening Gateways.

Welcome, and feel free to poke around!