ترغب بنشر مسار تعليمي؟ اضغط هنا

Introduction to mathematical logic - A problem solving course

71   0   0.0 ( 0 )
 نشر من قبل Arnold Miller
 تاريخ النشر 1996
  مجال البحث
والبحث باللغة English
 تأليف Arnold W. Miller




اسأل ChatGPT حول البحث

This is a set of 288 questions written for a Moore-style course in Mathematical Logic. I have used these (or some variation) four times in a beginning graduate course. Topics covered are: propositional logic axioms of ZFC wellorderings and equivalents of AC ordinal and cardinal arithmetic first order logic, and the compactness theorem Lowenheim-Skolem theorems Turing machines, Churchs Thesis completeness theorem and first incompleteness theorem undecidable theories second incompleteness theorem



قيم البحث

اقرأ أيضاً

265 - Zhaohua Luo 2011
In this paper, first-order logic is interpreted in the framework of universal algebra, using the clone theory developed in three previous papers. We first define the free clone T(L, C) of terms of a first order language L over a set C of parameters i n a standard way. The free right algebra F(L, C) of formulas over T(L, C) is then generated by atomic formulas. Structures for L over C are represented as perfect valuations of F(L, C), and theories of L are represented as filters of F(L). Finally Godels completeness theorem and first incompleteness theorem are stated as expected.
Many intellectual endeavors require mathematical problem solving, but this skill remains beyond the capabilities of computers. To measure this ability in machine learning models, we introduce MATH, a new dataset of 12,500 challenging competition math ematics problems. Each problem in MATH has a full step-by-step solution which can be used to teach models to generate answer derivations and explanations. To facilitate future research and increase accuracy on MATH, we also contribute a large auxiliary pretraining dataset which helps teach models the fundamentals of mathematics. Even though we are able to increase accuracy on MATH, our results show that accuracy remains relatively low, even with enormous Transformer models. Moreover, we find that simply increasing budgets and model parameter counts will be impractical for achieving strong mathematical reasoning if scaling trends continue. While scaling Transformers is automatically solving most other text-based tasks, scaling is not currently solving MATH. To have more traction on mathematical problem solving we will likely need new algorithmic advancements from the broader research community.
99 - Anton Freund 2021
These are the lecture notes of an introductory course on ordinal analysis. Our selection of topics is guided by the aim to give a complete and direct proof of a mathematical independence result: Kruskals theorem for binary trees is unprovable in cons ervative extensions of Peano arithmetic (note that much stronger results of this type are due to Harvey Friedman). Concerning prerequisites, we assume a solid introduction to mathematical logic but no specialized knowledge of proof theory. The material in these notes is intended for 14 lectures and 7 exercise sessions of 90 minutes each.
101 - Tarek Sayed Ahmed 2015
We take a long magical tour in algebraic logic, starting from classical results on neat embeddings due to Henkin, Monk and Tarski, all the way to recent results in algebraic logic using so--called rainbow constructions invented by Hirsch and Hodkinso n. Highlighting the connections with graph theory, model theory, and finite combinatorics, this article aspires to present topics of broad interest in a way that is hopefully accessible to a large audience. The paper has a survey character but it contains new approaches to old ones. We aspire to make our survey fairly comprehensive, at least in so far as Tarskian algebraic logic, specifically, the theory of cylindric algebras, is concerned. Other topics, such as abstract algebraic logic, modal logic and the so--called (central) finitizability problem in algebraic logic will be dealt with; the last in some detail. Rainbow constructions are used to solve problems adressing classes of cylindric--like algebras consisting of algebras having a neat embedding property. The hitherto obtained results generalize seminal results of Hirsch and Hodkinson on non--atom canonicity, non--first order definabiity and non--finite axiomatizability, proved for classes of representable cylindric algebras of finite dimension$>2$. We show that such results remain valid for cylindric algebras possesing relativized {it clique guarded} representations that are {it only locally} well behaved. The paper is written in a way that makes it accessible to non--specialists curious about the state of the art in Tarskian algebraic logic. Reaching the boundaries of current research, the paper also aspires to be informative to the practitioner, and even more, stimulates her/him to carry on further research in main stream algebraic logic.
We use the framework of reverse mathematics to address the question of, given a mathematical problem, whether or not it is easier to find an infinite partial solution than it is to find a complete solution. Following Flood, we say that a Ramsey-type variant of a problem is the problem with the same instances but whose solutions are the infinite partial solutions to the original problem. We study Ramsey-type variants of problems related to Konigs lemma, such as restrictions of Konigs lemma, Boolean satisfiability problems, and graph coloring problems. We find that sometimes the Ramsey-type variant of a problem is strictly easier than the original problem (as Flood showed with weak Konigs lemma) and that sometimes the Ramsey-type variant of a problem is equivalent to the original problem. We show that the Ramsey-type variant of weak Konigs lemma is robust in the sense of Montalban: it is equivalent to several perturbations of itself. We also clarify the relationship between Ramsey-type weak Konigs lemma and algorithmic randomness by showing that Ramsey-type weak weak Konigs lemma is equivalent to the problem of finding diagonally non-recursive functions and that these problems are strictly easier than Ramsey-type weak Konigs lemma. This answers a question of Flood.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا