{"product_id":"the-foundations-of-mathematics","title":"The Foundations of Mathematics","description":"\u003cp\u003eMathematical 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.\u003c\/p\u003e\u003cp\u003eThis book is designed for students who plan to specialize in logic, as well\u003c\/p\u003e\u003cp\u003eas 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\u003c\/p\u003e\u003cp\u003egraduate-level course.\u003c\/p\u003e\u003cp\u003eThere are three main chapters:\u003c\/p\u003e\u003cp\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.\u003c\/p\u003e\u003cp\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,\u003c\/p\u003e\u003cp\u003eelementary submodels, model completeness, and applications to algebra.\u003c\/p\u003e\u003cp\u003eThis chapter also continues the foundational issues begun in the set theory chapter.\u003c\/p\u003e\u003cp\u003eMathematics can now be viewed as formal proofs from ZFC.\u003c\/p\u003e\u003cp\u003eAlso, model theory leads to models of set theory.\u003c\/p\u003e\u003cp\u003eThis includes a discussion of absoluteness, and an analysis of models such as H(?) and R(?).\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":67647279530351,"sku":"9781904987147","price":206.56,"currency_code":"BRL","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0921\/9384\/9711\/files\/1904987141.jpg?v=1790890342","url":"https:\/\/internacional.umlivro.com.br\/products\/the-foundations-of-mathematics","provider":"UmLivro Internacional","version":"1.0","type":"link"}