Replacement Versus Collection and Related Topics in Constructive Zermelo-Fraenkel Set Theory
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Specifying Verified X86 Software from Scratch
Specifying verified x86 software from scratch Mario Carneiro Carnegie Mellon University, Pittsburgh, PA, USA [email protected] Abstract We present a simple framework for specifying and proving facts about the input/output behavior of ELF binary files on the x86-64 architecture. A strong emphasis has been placed on simplicity at all levels: the specification says only what it needs to about the target executable, the specification is performed inside a simple logic (equivalent to first-order Peano Arithmetic), and the verification language and proof checker are custom-designed to have only what is necessary to perform efficient general purpose verification. This forms a part of the Metamath Zero project, to build a minimal verifier that is capable of verifying its own binary. In this paper, we will present the specification of the dynamic semantics of x86 machine code, together with enough information about Linux system calls to perform simple IO. 2012 ACM Subject Classification Theory of computation → Program verification; Theory of com- putation → Abstract machines Keywords and phrases x86-64, ISA specification, Metamath Zero, self-verification Digital Object Identifier 10.4230/LIPIcs.ITP.2019.19 Supplement Material The formalization is a part of the Metamath Zero project at https://github. com/digama0/mm0. Funding This material is based upon work supported by AFOSR grant FA9550-18-1-0120 and a grant from the Sloan Foundation. Acknowledgements I would like to thank my advisor Jeremy Avigad for his support and encourage- ment, and for his reviews of early drafts of this work. 1 Introduction The field of software verification is on the verge of a breakthrough. -
The Metamathematics of Putnam's Model-Theoretic Arguments
The Metamathematics of Putnam's Model-Theoretic Arguments Tim Button Abstract. Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted mod- els. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I exam- ine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to meta- mathematical challenges. Copyright notice. This paper is due to appear in Erkenntnis. This is a pre-print, and may be subject to minor changes. The authoritative version should be obtained from Erkenntnis, once it has been published. Hilary Putnam famously attempted to use model theory to draw metaphys- ical conclusions. Specifically, he attacked metaphysical realism, a position characterised by the following credo: [T]he world consists of a fixed totality of mind-independent objects. (Putnam 1981, p. 49; cf. 1978, p. 125). Truth involves some sort of correspondence relation between words or thought-signs and external things and sets of things. (1981, p. 49; cf. 1989, p. 214) [W]hat is epistemically most justifiable to believe may nonetheless be false. (1980, p. 473; cf. 1978, p. 125) To sum up these claims, Putnam characterised metaphysical realism as an \externalist perspective" whose \favorite point of view is a God's Eye point of view" (1981, p. 49). Putnam sought to show that this externalist perspective is deeply untenable. To this end, he treated correspondence in terms of model-theoretic satisfaction. -
Mathematics in the Computer
Mathematics in the Computer Mario Carneiro Carnegie Mellon University April 26, 2021 1 / 31 Who am I? I PhD student in Logic at CMU I Proof engineering since 2013 I Metamath (maintainer) I Lean 3 (maintainer) I Dabbled in Isabelle, HOL Light, Coq, Mizar I Metamath Zero (author) Github: digama0 I Proved 37 of Freek’s 100 theorems list in Zulip: Mario Carneiro Metamath I Lots of library code in set.mm and mathlib I Say hi at https://leanprover.zulipchat.com 2 / 31 I Found Metamath via a random internet search I they already formalized half of the book! I .! . and there is some stuff on cofinality they don’t have yet, maybe I can help I Got involved, did it as a hobby for a few years I Got a job as an android developer, kept on the hobby I Norm Megill suggested that I submit to a (Mizar) conference, it went well I Met Leo de Moura (Lean author) at a conference, he got me in touch with Jeremy Avigad (my current advisor) I Now I’m a PhD at CMU philosophy! How I got involved in formalization I Undergraduate at Ohio State University I Math, CS, Physics I Reading Takeuti & Zaring, Axiomatic Set Theory 3 / 31 I they already formalized half of the book! I .! . and there is some stuff on cofinality they don’t have yet, maybe I can help I Got involved, did it as a hobby for a few years I Got a job as an android developer, kept on the hobby I Norm Megill suggested that I submit to a (Mizar) conference, it went well I Met Leo de Moura (Lean author) at a conference, he got me in touch with Jeremy Avigad (my current advisor) I Now I’m a PhD at CMU philosophy! How I got involved in formalization I Undergraduate at Ohio State University I Math, CS, Physics I Reading Takeuti & Zaring, Axiomatic Set Theory I Found Metamath via a random internet search 3 / 31 I . -
The Metamath Proof Language
The Metamath Proof Language Norman Megill May 9, 2014 1 Metamath development server Overview of Metamath • Very simple language: substitution is the only basic rule • Very small verifier (≈300 lines code) • Fast proof verification (6 sec for ≈18000 proofs) • All axioms (including logic) are specified by user • Formal proofs are complete and transparent, with no hidden implicit steps 2 Goals Simplest possible framework that can express and verify (essentially) all of mathematics with absolute rigor Permanent archive of hand-crafted formal proofs Elimination of uncertainty of proof correctness Exposure of missing steps in informal proofs to any level of detail desired Non-goals (at this time) Automated theorem proving Practical proof-finding assistant for working mathematicians 3 sophistication × × × × × others Metamath × transparency (Ficticious conceptual chart) 4 Contributors David Abernethy David Harvey Rodolfo Medina Stefan Allan Jeremy Henty Mel L. O'Cat Juha Arpiainen Jeff Hoffman Jason Orendorff Jonathan Ben-Naim Szymon Jaroszewicz Josh Purinton Gregory Bush Wolf Lammen Steve Rodriguez Mario Carneiro G´erard Lang Andrew Salmon Paul Chapman Raph Levien Alan Sare Scott Fenton Fr´ed´ericLin´e Eric Schmidt Jeffrey Hankins Roy F. Longton David A. Wheeler Anthony Hart Jeff Madsen 5 Examples of axiom systems expressible with Metamath (Blue means used by the set.mm database) • Intuitionistic, classical, paraconsistent, relevance, quantum propositional logics • Free or standard first-order logic with equality; modal and provability logics • NBG, ZF, NF set theory, with AC, GCH, inaccessible and other large cardinal axioms Axiom schemes are exact logical equivalents to textbook counterparts. All theorems can be instantly traced back to what axioms they use. -
Constructing a Categorical Framework of Metamathematical Comparison Between Deductive Systems of Logic
Bard College Bard Digital Commons Senior Projects Spring 2016 Bard Undergraduate Senior Projects Spring 2016 Constructing a Categorical Framework of Metamathematical Comparison Between Deductive Systems of Logic Alex Gabriel Goodlad Bard College, [email protected] Follow this and additional works at: https://digitalcommons.bard.edu/senproj_s2016 Part of the Logic and Foundations Commons This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License. Recommended Citation Goodlad, Alex Gabriel, "Constructing a Categorical Framework of Metamathematical Comparison Between Deductive Systems of Logic" (2016). Senior Projects Spring 2016. 137. https://digitalcommons.bard.edu/senproj_s2016/137 This Open Access work is protected by copyright and/or related rights. It has been provided to you by Bard College's Stevenson Library with permission from the rights-holder(s). You are free to use this work in any way that is permitted by the copyright and related rights. For other uses you need to obtain permission from the rights- holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/or on the work itself. For more information, please contact [email protected]. Constructing a Categorical Framework of Metamathematical Comparison Between Deductive Systems of Logic A Senior Project submitted to The Division of Science, Mathematics, and Computing of Bard College by Alex Goodlad Annandale-on-Hudson, New York May, 2016 Abstract The topic of this paper in a broad phrase is \proof theory". It tries to theorize the general notion of \proving" something using rigorous definitions, inspired by previous less general theories. The purpose for being this general is to eventually establish a rigorous framework that can bridge the gap when interrelating different logical systems, particularly ones that have not been as well defined rigorously, such as sequent calculus. -
The Realm of Ordinal Analysis
The Realm of Ordinal Analysis Michael Rathjen Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, United Kingdom Denn die Pioniere der Mathematik hatten sich von gewissen Grundlagen brauchbare Vorstellungen gemacht, aus denen sich Schl¨usse,Rechnungsarten, Resultate ergaben, deren bem¨achtigten sich die Physiker, um neue Ergebnisse zu erhalten, und endlich kamen die Techniker, nahmen oft bloß die Resultate, setzten neue Rechnungen darauf und es entstanden Maschinen. Und pl¨otzlich, nachdem alles in sch¨onste Existenz gebracht war, kamen die Mathematiker - jene, die ganz innen herumgr¨ubeln, - darauf, daß etwas in den Grundlagen der ganzen Sache absolut nicht in Ordnung zu bringen sei; tats¨achlich, sie sahen zuunterst nach und fanden, daß das ganze Geb¨audein der Luft stehe. Aber die Maschinen liefen! Man muß daraufhin annehmen, daß unser Dasein bleicher Spuk ist; wir leben es, aber eigentlich nur auf Grund eines Irrtums, ohne den es nicht entstanden w¨are. ROBERT MUSIL: Der mathematische Mensch (1913) 1 Introduction A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their ‘rank’ or ‘complexity’ in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of ‘proof the- oretic ordinals’ to theories, gauging their ‘consistency strength’ and ‘computational power’. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen’s head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories. -
On Relating Theories: Proof-Theoretical Reduction
This is a repository copy of On Relating Theories: Proof-Theoretical Reduction. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/147184/ Version: Accepted Version Proceedings Paper: Rathjen, M and Toppel, M (2019) On Relating Theories: Proof-Theoretical Reduction. In: Centrone, S, Negri, S, Sarikaya, D and Schuster, PM, (eds.) Mathesis Universalis, Computability and Proof. Humboldt-Kolleg: Proof Theory as Mathesis Universalis, 24-28 Jul 2017, Loveno di Menaggio (Como), Italy. Springer , pp. 311-331. ISBN 978-3-030-20446-4 https://doi.org/10.1007/978-3-030-20447-1_16 © Springer Nature Switzerland AG 2019. This is an author produced version of a paper published in Mathesis Universalis, Computability and Proof. Uploaded in accordance with the publisher's self-archiving policy. Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ On relating theories: Proof-theoretical reduction Michael Rathjen and Michael Toppel Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, United Kingdom E-Mail: [email protected] Abstract The notion of proof-theoretical or finitistic reduction of one theory to another has a long tradition. -
Arxiv:2005.03089V1 [Cs.LO] 5 May 2020 Abstract ??? Accepted: / ??? Received: Optrsine a Erlangen-N¨Urnberg E.G
Noname manuscript No. (will be inserted by the editor) Experiences from Exporting Major Proof Assistant Libraries Michael Kohlhase · Florian Rabe Received: ??? / Accepted: ??? Abstract The interoperability of proof assistants and the integration of their li- braries is a highly valued but elusive goal in the field of theorem proving. As a preparatory step, in previous work, we translated the libraries of multiple proof assistants, specifically the ones of Coq, HOL Light, IMPS, Isabelle, Mizar, and PVS into a universal format: OMDoc/MMT. Each translation presented tremendous theoretical, technical, and social chal- lenges, some universal and some system-specific, some solvable and some still open. In this paper, we survey these challenges and compare and evaluate the solutions we chose. We believe similar library translations will be an essential part of any future system interoperability solution and our experiences will prove valuable to others undertaking such efforts. 1 Introduction Motivation The hugely influential QED manifesto [2] of 1994 urged the automated reasoning community to work towards a universal, computer-based database of all mathematical knowledge, strictly formalized in logic and supported by proofs that can be checked mechanically. The QED database was intended as a communal re- source that would guide research and allow the evaluation of automated reasoning tools and systems. This database was never realized, but the interoperability of proof assistants and the integration of their libraries has remained a highly valued but elusive goal. This is despite the large and growing need for more integrated and easily reusable libraries of formalizations. For example, the Flyspeck project [20] build a formal proof of the Kepler conjecture in HOL Light. -
Generating Custom Set Theories with Non-Set Structured Objects
EasyChair Preprint № 5663 Generating Custom Set Theories with Non-Set Structured Objects Ciarán Dunne, Joe Wells and Fairouz Kamareddine EasyChair preprints are intended for rapid dissemination of research results and are integrated with the rest of EasyChair. June 1, 2021 Generating Custom Set Theories with Non-Set Structured Objects Ciar´anDunne, J. B. Wells, Fairouz Kamareddine Heriot-Watt University Abstract. Set theory has long been viewed as a foundation of math- ematics, is pervasive in mathematical culture, and is explicitly used by much written mathematics. Because arrangements of sets can represent a vast multitude of mathematical objects, in most set theories every object is a set. This causes confusion and adds difficulties to formalising math- ematics in set theory. We wish to have set theory's features while also having many mathematical objects not be sets. A generalized set theory (GST) is a theory that has pure sets and may also have non-sets that can have internal structure and impure sets that mix sets and non-sets. This paper provides a GST-building framework. We show example GSTs that have sets and also (1) non-set ordered pairs, (2) non-set natural numbers, (3) a non-set exception object that can not be inside another object, and (4) modular combinations of these features. We show how to axiomatize GSTs and how to build models for GSTs in other GSTs. 1 Introduction Set Theory as a Foundation of Mathematics. Set theories like Zermelo- Fraenkel (ZF), and closely related set theories like ZFC and Tarski-Grothendieck (TG), play many important roles in mathematics. -
The Birth of Model Theory
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 47, Number 1, January 2010, Pages 177–185 S 0273-0979(09)01275-0 Article electronically published on September 8, 2009 The birth of model theory: L¨owenheim’s theory in the frame of the theory of relatives, by Calixto Badesa, Princeton University Press, Princeton, NJ, 2004, xiv+240 pp., ISBN 978-0-691-05853-5, hardcover, US$64.00 What do the theorems of G¨odel–Deligne, Chevalley–Tarski, Ax–Grothendieck, Tarski–Seidenberg, and Weil–Hrushovski have in common? And what do they have to do with the book under review? Each of these theorems was proven by techniques in a particular mathematical area and by model theoretic methods. In fact, these model theoretic methods often show a pattern that extends across these areas. What are model theoretic methods? Model theory is the activity of a “self- conscious” mathematician. This mathematician distinguishes an object language (syntax) and a class of structures for this language and “definable” subsets of those structures (semantics). Semantics provides an interpretation of inscriptions in the formal language in the appropriate structures. At its most basic level this allows the recognition that syntactic transformations can clarify the description of the same set of numbers. Thus, x2 − 3x<−6 is rewritten as x<−2orx>3; both formulas define the same set of points if they are interpreted in the real numbers. After clarifying these fundamental notions, we give an anachronistic survey of three themes of twentieth century model theory: the study of a) properties of first order logic, b) specific first order theories, and c) classification of first order theories. -
Models for Metamath
Models for Metamath Mario Carneiro Carnegie Mellon University, Pittsburgh PA, USA Abstract. Although some work has been done on the metamathematics of Metamath, there has not been a clear definition of a model for a Meta- math formal system. We define the collection of models of an arbitrary Metamath formal system, both for tree-based and string-based represen- tations. This definition is demonstrated with examples for propositional calculus, ZFC set theory with classes, and Hofstadter's MIU system, with applications for proving that statements are not provable, showing con- sistency of the main Metamath database (assuming ZFC has a model), developing new independence proofs, and proving a form of G¨odel'scom- pleteness theorem. Keywords: Metamath · Model theory · formal proof · consistency · ZFC · Mathematical logic 1 Introduction Metamath is a proof language, developed in 1992, on the principle of minimizing the foundational logic to as little as possible [1]. An expression in Metamath is a string of constants and variables headed by a constant called the expression's \typecode". The variables are typed and can be substituted for expressions with the same typecode. See x 2.1 for a precise definition of a formal system, which mirrors the specification of the .mm file format itself. The logic on which Metamath is based was originally defined by Tarski in [2]. Notably, this involves a notion of \direct" or \non-capturing" substitution, which means that no α-renaming occurs during a substitution for a variable. Instead, this is replaced by a \distinct variable" condition saying that certain substitu- tions are not valid if they contain a certain variable (regardless of whether the variable is free or not|Metamath doesn't know what a free variable is). -
Constructive Zermelo-Fraenkel Set Theory, Power Set, and the Calculus of Constructions
This is a repository copy of Constructive zermelo-fraenkel set theory, power set, and the calculus of constructions. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/75182/ Book Section: Rathjen, M (2012) Constructive zermelo-fraenkel set theory, power set, and the calculus of constructions. In: Dybjer, P, Lindström, S, Palmgren, E and Sundholm, G, (eds.) Epistemology versus ontology: Essays on the philosophy and foundations of mathematics in honour of Per Martin-Löf. Logic, Epistemology, and the Unity of Science, 27 . Springer , Dordrecht, Netherlands , 313 - 349. ISBN 9789400744356 https://doi.org/10.1007/978-94-007-4435-6 Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Constructive Zermelo-Fraenkel Set Theory, Power Set, and the Calculus of Constructions Michael Rathjen∗ Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, United Kingdom [email protected] May 4, 2012 Abstract Full intuitionistic Zermelo-Fraenkel set theory, IZF, is obtained from constructive Zermelo- Fraenkel set theory, CZF, by adding the full separation axiom scheme and the power set axiom.