{"product_id":"9781904987147","title":"The Foundations Of Mathematics","description":"Mathematical logic grew out of philosophical questions regarding the foundations of mathematics, but logic has now outgrown its philosophical roots, and has become an integral part of mathematics in general.\u003cbr\u003eThis book is designed for students who plan to specialize in logic, as well as for those who are interested in the applications of logic to other areas of mathematics. Used as a text, it could form the basis of a beginning graduate-level course.\u003cp\u003eThere are three main chapters:\u003cbr\u003eSet Theory, Model Theory, and Recursion Theory.\u003c\/p\u003e\u003cp\u003eThe Set Theory chapter describes the set-theoretic foundations of all of mathematics, based on the ZFC axioms.\u003cbr\u003eIt also covers technical results about the Axiom of Choice, well-orderings, and the theory of uncountable cardinals.\u003c\/p\u003e\u003cp\u003eThe Model Theory chapter discusses predicate logic and formal proofs, and covers the Completeness, Compactness, and Löwenheim-Skolem Theorems,\u003cbr\u003eelementary submodels, model completeness, and applications to algebra.\u003cbr\u003eThis chapter also continues the foundational issues begun in the set theory chapter.\u003cbr\u003eMathematics can now be viewed as formal proofs from ZFC.\u003cbr\u003eAlso, model theory leads to models of set theory.\u003cbr\u003eThis includes a discussion of absoluteness, and an analysis of models such as H(κ) and R(γ).\u003c\/p\u003e\u003cp\u003eThe Recursion Theory chapter develops some basic facts about computable functions, and uses them to prove a number of results of foundational importance; in particular, Church's theorem on the undecidability of logical consequence, the incompleteness theorems of Gödel, and Tarski's theorem on the non-definability of truth.\u003c\/p\u003e","brand":"College Publications","offers":[{"title":"Default Title","offer_id":47039900647664,"sku":"9781904987147","price":24.75,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0737\/7593\/9824\/files\/9781904987147_p0.jpg?v=1763615466","url":"https:\/\/shop-qa.barnesandnoble.com\/products\/9781904987147","provider":"Barnes \u0026 Noble (DEV)","version":"1.0","type":"link"}