دانلود کتاب قضایای گودل و اصول موضوعه زرملوس

This book provides a self-contained introduction to the foundations of mathematics, where self-contained means that we assume as little prerequisites as possible. One such assumption is the notion of finiteness, which cannot be defined in mathematics. The firm foundation of mathematics we provide is based on logic and models. In particular, it is based on Hilbert’s axiomatisation of formal logic (including the notion of formal proofs), and on the notion of models of mathematical theories. On this basis, we first prove G¨odel’s Completeness Theorem and G¨odel’s Incompleteness Theorems, and then we introduce Zermelo’s Axioms of Set Theory. On the one hand, G¨odel’s Theorems set the framework within which mathematics takes place. On the other hand, using the example of Analysis, we shall see how mathematics can be developed in a model of Set Theory. So, G¨odel’s Theorems and Zermelo’s Axioms are indeed a firm foundation of mathematics. The book consists of four parts. The first part is an introduction to First- Order Logic from scratch. Starting with a set of symbols, the basic concepts of formal proofs and models are developed, where special care is given to the notion of finiteness. The second part is concerned with G¨odel’s Completeness Theorem. Our proof follows Henkin’s construction [23]. However, we modified Henkin’s construction in order to work just with potentially infinite sets and to avoid the use of actually infinite sets. Even though Henkin’s construction works also for uncountable signatures, we prove the general Completeness Theorem with an ultraproduct construction, using Loˇs’s Theorem. After a preliminary chapter on countable models of Peano Arithmetic, the third part is mainly concerned with G¨odel’s Incompleteness Theorems, which will be proved from scratch (i.e., purely from the axioms of Peano Arithmetic) without any use of Recursion Theory. In Chapter 10 and 12 some weaker theories of arithmetic are investigated. In particular, in Chapter 10 it is shown that G¨odel’s First Incompleteness Theorem also applies for Robinson Arithmetic and in Chapter 12 it is shown that Presburger Arithmetic is complete. In the last part we present first Zermelo’s axioms of Set Theory, including the Axiom of Choice. Then we discuss the consistency of this axiomatic system and provide standard and non-standard models of Set Theory (including G¨odel’s model L). After introducing the construction of models with ultraproducts, we prove the Completeness Theorem for uncountable signatures as well the the L¨owenheim-Skolem Theorems. In the last two chapters, we construct several standard and non-standard models of Peano Arithmetic and of the real numbers, and give a brief introduction to Non- Standard Analysis.