The omega-rule interpretation of transfinite provability logic

A

Mathematical Logic: Proof theory, Constructive Mathematics

The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit

The Structure of Models of Second-order Set Theories

This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of T-realizations of a fixed countable model of ZFC, where T is a reasonable second-order set theory such as GBC or KM, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve the existence/nonexistence of upper bounds, at least for finite partial orders. Second I generalize some constructions of Marek and Mostowski from KM to weaker theories. They showed that every model of KM plus the Class Collection schema “unrolls” to a model of ZFC− with a largest cardinal. I calculate the theories of the unrolling for a variety of second-order set theories, going as weak as GBC + ETR. I also show that being T-realizable goes down to submodels for a broad selection of second-order set theories T. Third, I show that there is a hierarchy of transfinite recursion principles ranging in strength from GBC to KM. This hierarchy is ordered first by the complexity of the properties allowed in the recursions and second by the allowed heights of the recursions. Fourth, I investigate the question of which second-order set theories have least models. I show that strong theories—such as KM or Π11-CA—do not have least transitive models while weaker theories—from GBC to GBC + ETROrd —do have least transitive models

The Structure of Models of Second-order Set Theories

This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of $T$-realizations of a fixed countable model of $\mathsf{ZFC}$, where $T$ is a reasonable second-order set theory such as $\mathsf{GBC}$ or $\mathsf{KM}$, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve the existence/nonexistence of upper bounds, at least for finite partial orders. Second I generalize some constructions of Marek and Mostowski from $\mathsf{KM}$ to weaker theories. They showed that every model of $\mathsf{KM}$ plus the Class Collection schema "unrolls" to a model of $\mathsf{ZFC}^-$ with a largest cardinal. I calculate the theories of the unrolling for a variety of second-order set theories, going as weak as $\mathsf{GBC} + \mathsf{ETR}$. I also show that being $T$-realizable goes down to submodels for a broad selection of second-order set theories $T$. Third, I show that there is a hierarchy of transfinite recursion principles ranging in strength from $\mathsf{GBC}$ to $\mathsf{KM}$. This hierarchy is ordered first by the complexity of the properties allowed in the recursions and second by the allowed heights of the recursions. Fourth, I investigate the question of which second-order set theories have least models. I show that strong theories---such as $\mathsf{KM}$ or $\Pi^1_1\text{-}\mathsf{CA}$---do not have least transitive models while weaker theories---from $\mathsf{GBC}$ to $\mathsf{GBC} + \mathsf{ETR}_\mathrm{Ord}$---do have least transitive models.Comment: This is my PhD dissertatio

The Implicit Commitment of Arithmetical Theories and Its Semantic Core

According to the implicit commitment thesis, once accepting a mathematical formal system S, one is implicitly committed to additional resources not immediately available in S. Traditionally, this thesis has been understood as entailing that, in accepting S, we are bound to accept reflection principles for S and therefore claims in the language of S that are not derivable in S itself. It has recently become clear, however, that such reading of the implicit commitment thesis cannot be compatible with well-established positions in the foundations of mathematics which consider a specific theory S as self-justifying and doubt the legitimacy of any principle that is not derivable in S: examples are Tait’s finitism and the role played in it by Primitive Recursive Arithmetic, Isaacson’s thesis and Peano Arithmetic, Nelson’s ultrafinitism and sub-exponential arithmetical systems. This casts doubts on the very adequacy of the implicit commitment thesis for arithmetical theories. In the paper we show that such foundational standpoints are nonetheless compatible with the implicit commitment thesis. We also show that they can even be compatible with genuine soundness extensions of S with suitable form of reflection. The analysis we propose is as follows: when accepting a system S, we are bound to accept a fixed set of principles extending S and expressing minimal soundness requirements for S, such as the fact that the non-logical axioms of S are true. We call this invariant component the semantic core of implicit commitment. But there is also a variable component of implicit commitment that crucially depends on the justification given for our acceptance of S in which, for instance, may or may not appear (proof-theoretic) reflection principles for S. We claim that the proposed framework regulates in a natural and uniform way our acceptance of different arithmetical theories

Kant, infinity and his first antinomy

Kant's antinomies are exercises designed to illustrate the limits of human reasoning. He skillfully juxtaposes pairs of arguments and exposes the dangerous propensity for human reasoning to stretch beyond the conditioned and into the transcendental ideas of the unconditional. Kant believes this is a natural process and affirms the limits of pure reason in so much as they should prevent us from believing that we can truly know anything about the unconditional. His first antimony addresses the possibility of a beginning in time or no beginning in time. This thesis will focus on this first antinomy and critically assesses it in set theoretic terms. It is this author's belief that the mathematical nuances of infinite sets and the understanding of mathematical objects bear relevance to the proper interpretation of this antinomy. Ultimately, this composition will illustrate that Kant's argument in the first antinomy is flawed because it fails to account for infinite bounded sets and a conceptualization of the infinite as a mathematical object of reason

Proof theory and Martin-Löf Type Theory

In this article an overview over the work of the author on developing proof theoretic strong extensions of Martin-Loef Type Theory including precise proof theoretic bounds is given. It presents the first publication of the proof theoretically strongest known extensions of Martin-Loef Type Theory, namely the hyper-Mahlo Universe, the hyper-alpha-Mahlo universe, the autononomous Mahlo universe and the Pi_3-reflecting universe. This is part of a proof theoretic program in developing proof theoretic as strong as possible constructive theories in order to obtain a constructive underpinning of strong classical theories with a full proof theoretic analysis

esys-Escript User’s Guide: Solving Partial Differential Equations with Escript and Finley Release - 3.3.1 (r4302)

esys.escript is a python-based environment for implementing mathematical models, in particular those based on coupled, non-linear, time-dependent partial differential equations. It consists of five major components • esys.escript core library • finite element solver esys.finley (which uses fast vendor-supplied solvers or our paso linear solver library) • the meshing interface esys.pycad • a model library. • an inversion library. The current version supports parallelization through both MPI for distributed memory and OpenMP for distributed shared memory. In this release there are a number of small changes which are not backwards compatible. Please see Appendix B to see if your scripts will be affected. If you use this software in your research, then we would appreciate (but do not require) a citation. Some relevant references can be found in Appendix D. For Python3 support, see Appendix E