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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). https://doi.org/10.1007/s11225-019-09860-7.  pdf, arXiv

 

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.

https://nylogic.github.io/set-theory-seminar/2019/05/17/generalized-cohen-iterations.html

 

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.

Publications

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). https://doi.org/10.1007/s11225-019-09860-7.  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.  pdfarXiv

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.

Dissertation.