# Elements Of Set Theory And Mathematical Logic Pdf

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*Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.*

## Set Theory Papers

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics , the foundations of mathematics , and theoretical computer science. Mathematical logic is often divided into the fields of set theory , model theory , recursion theory , and proof theory. These areas share basic results on logic, particularly first-order logic , and definability. In computer science particularly in the ACM Classification mathematical logic encompasses additional topics not detailed in this article; see Logic in computer science for those.

Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry , arithmetic , and analysis. In the early 20th century it was shaped by David Hilbert 's program to prove the consistency of foundational theories. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory.

Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems as in reverse mathematics rather than trying to find theories in which all of mathematics can be developed. The Handbook of Mathematical Logic [2] in makes a rough division of contemporary mathematical logic into four areas:.

Each area has a distinct focus, although many techniques and results are shared among multiple areas. The borderlines amongst these fields, and the lines separating mathematical logic and other fields of mathematics, are not always sharp.

The method of forcing is employed in set theory, model theory, and recursion theory, as well as in the study of intuitionistic mathematics. The mathematical field of category theory uses many formal axiomatic methods, and includes the study of categorical logic , but category theory is not ordinarily considered a subfield of mathematical logic.

Because of its applicability in diverse fields of mathematics, mathematicians including Saunders Mac Lane have proposed category theory as a foundational system for mathematics, independent of set theory. These foundations use toposes , which resemble generalized models of set theory that may employ classical or nonclassical logic. The first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics.

Theories of logic were developed in many cultures in history, including China , India , Greece and the Islamic world. Greek methods, particularly Aristotelian logic or term logic as found in the Organon , found wide application and acceptance in Western science and mathematics for millennia.

In 18th-century Europe, attempts to treat the operations of formal logic in a symbolic or algebraic way had been made by philosophical mathematicians including Leibniz and Lambert , but their labors remained isolated and little known. In the middle of the nineteenth century, George Boole and then Augustus De Morgan presented systematic mathematical treatments of logic.

Charles Sanders Peirce later built upon the work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from to Gottlob Frege presented an independent development of logic with quantifiers in his Begriffsschrift , published in , a work generally considered as marking a turning point in the history of logic. Frege's work remained obscure, however, until Bertrand Russell began to promote it near the turn of the century. The two-dimensional notation Frege developed was never widely adopted and is unused in contemporary texts.

This work summarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to symbolic logic as it was understood at the end of the 19th century. Concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for fundamental areas of mathematics such as arithmetic, analysis, and geometry.

In logic, the term arithmetic refers to the theory of the natural numbers. Peano was unaware of Frege's work at the time. Around the same time Richard Dedekind showed that the natural numbers are uniquely characterized by their induction properties.

Dedekind proposed a different characterization, which lacked the formal logical character of Peano's axioms. Dedekind's work, however, proved theorems inaccessible in Peano's system, including the uniqueness of the set of natural numbers up to isomorphism and the recursive definitions of addition and multiplication from the successor function and mathematical induction.

In the midth century, flaws in Euclid's axioms for geometry became known Katz , p. In addition to the independence of the parallel postulate , established by Nikolai Lobachevsky in Lobachevsky , mathematicians discovered that certain theorems taken for granted by Euclid were not in fact provable from his axioms. Among these is the theorem that a line contains at least two points, or that circles of the same radius whose centers are separated by that radius must intersect.

Hilbert developed a complete set of axioms for geometry , building on previous work by Pasch The success in axiomatizing geometry motivated Hilbert to seek complete axiomatizations of other areas of mathematics, such as the natural numbers and the real line.

This would prove to be a major area of research in the first half of the 20th century. The 19th century saw great advances in the theory of real analysis , including theories of convergence of functions and Fourier series. Mathematicians such as Karl Weierstrass began to construct functions that stretched intuition, such as nowhere-differentiable continuous functions. Previous conceptions of a function as a rule for computation, or a smooth graph, were no longer adequate.

Weierstrass began to advocate the arithmetization of analysis , which sought to axiomatize analysis using properties of the natural numbers. Cauchy in defined continuity in terms of infinitesimals see Cours d'Analyse, page In , Dedekind proposed a definition of the real numbers in terms of Dedekind cuts of rational numbers Dedekind , a definition still employed in contemporary texts. Georg Cantor developed the fundamental concepts of infinite set theory.

His early results developed the theory of cardinality and proved that the reals and the natural numbers have different cardinalities Cantor Over the next twenty years, Cantor developed a theory of transfinite numbers in a series of publications. In , he published a new proof of the uncountability of the real numbers that introduced the diagonal argument , and used this method to prove Cantor's theorem that no set can have the same cardinality as its powerset. Cantor believed that every set could be well-ordered , but was unable to produce a proof for this result, leaving it as an open problem in Katz , p.

In the early decades of the 20th century, the main areas of study were set theory and formal logic. The discovery of paradoxes in informal set theory caused some to wonder whether mathematics itself is inconsistent, and to look for proofs of consistency. In , Hilbert posed a famous list of 23 problems for the next century.

The first two of these were to resolve the continuum hypothesis and prove the consistency of elementary arithmetic, respectively; the tenth was to produce a method that could decide whether a multivariate polynomial equation over the integers has a solution. Subsequent work to resolve these problems shaped the direction of mathematical logic, as did the effort to resolve Hilbert's Entscheidungsproblem , posed in This problem asked for a procedure that would decide, given a formalized mathematical statement, whether the statement is true or false.

Ernst Zermelo gave a proof that every set could be well-ordered , a result Georg Cantor had been unable to obtain. To achieve the proof, Zermelo introduced the axiom of choice , which drew heated debate and research among mathematicians and the pioneers of set theory. The immediate criticism of the method led Zermelo to publish a second exposition of his result, directly addressing criticisms of his proof Zermelo a. This paper led to the general acceptance of the axiom of choice in the mathematics community.

Skepticism about the axiom of choice was reinforced by recently discovered paradoxes in naive set theory. Cesare Burali-Forti was the first to state a paradox: the Burali-Forti paradox shows that the collection of all ordinal numbers cannot form a set. Very soon thereafter, Bertrand Russell discovered Russell's paradox in , and Jules Richard discovered Richard's paradox. Zermelo b provided the first set of axioms for set theory. These axioms, together with the additional axiom of replacement proposed by Abraham Fraenkel , are now called Zermelo—Fraenkel set theory ZF.

Zermelo's axioms incorporated the principle of limitation of size to avoid Russell's paradox. This seminal work developed the theory of functions and cardinality in a completely formal framework of type theory , which Russell and Whitehead developed in an effort to avoid the paradoxes.

Fraenkel proved that the axiom of choice cannot be proved from the axioms of Zermelo's set theory with urelements. Later work by Paul Cohen showed that the addition of urelements is not needed, and the axiom of choice is unprovable in ZF. Cohen's proof developed the method of forcing , which is now an important tool for establishing independence results in set theory.

Skolem realized that this theorem would apply to first-order formalizations of set theory, and that it implies any such formalization has a countable model.

This counterintuitive fact became known as Skolem's paradox. These results helped establish first-order logic as the dominant logic used by mathematicians. It showed the impossibility of providing a consistency proof of arithmetic within any formal theory of arithmetic. Hilbert, however, did not acknowledge the importance of the incompleteness theorem for some time.

This leaves open the possibility of consistency proofs that cannot be formalized within the system they consider. Gentzen proved the consistency of arithmetic using a finitistic system together with a principle of transfinite induction. Gentzen's result introduced the ideas of cut elimination and proof-theoretic ordinals , which became key tools in proof theory. Alfred Tarski developed the basics of model theory. These texts, written in an austere and axiomatic style, emphasized rigorous presentation and set-theoretic foundations.

Terminology coined by these texts, such as the words bijection , injection , and surjection , and the set-theoretic foundations the texts employed, were widely adopted throughout mathematics. Kleene introduced the concepts of relative computability, foreshadowed by Turing , and the arithmetical hierarchy. Kleene later generalized recursion theory to higher-order functionals.

Kleene and Georg Kreisel studied formal versions of intuitionistic mathematics, particularly in the context of proof theory. At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems.

These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language. The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties.

First-order logic is a particular formal system of logic. Its syntax involves only finite expressions as well-formed formulas , while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse.

Early results from formal logic established limitations of first-order logic. This shows that it is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism. As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark. It shows that if a particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms.

It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset.

The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory , and they are a key reason for the prominence of first-order logic in mathematics. The first incompleteness theorem states that for any consistent, effectively given defined below logical system that is capable of interpreting arithmetic, there exists a statement that is true in the sense that it holds for the natural numbers but not provable within that logical system and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system.

Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong. The second incompleteness theorem states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be reached.

Many logics besides first-order logic are studied.

## Discrete Mathematics - Sets

Set theory is the mathematical theory of well-determined collections, called sets , of objects that are called members , or elements , of the set. Pure set theory deals exclusively with sets, so the only sets under consideration are those whose members are also sets. The theory of the hereditarily-finite sets, namely those finite sets whose elements are also finite sets, the elements of which are also finite, and so on, is formally equivalent to arithmetic. So, the essence of set theory is the study of infinite sets, and therefore it can be defined as the mathematical theory of the actual—as opposed to potential—infinite. The notion of set is so simple that it is usually introduced informally, and regarded as self-evident.

Home About My account Contact Us. For our purposes, it will sufce to [You may assume any standard results not concerning ordinal e xponentiation. The subjects of register machines and random access machines have been dropped from Section 5. Justify your answers. Logic andSet Theory Lectured by I. Calini, E.

Like logic, the subject of sets is rich and interesting for its own sake. We will need only a few facts about sets and techniques for dealing with them, which we set out in this section and the next. We will return to sets as an object of study in chapters 4 and 5. A set is a collection of objects; any one of the objects in a set is called a member or an element of the set. Some sets occur so frequently that there are standard names and symbols for them. There is a natural relationship between sets and logic.

ELEMENTS OF MATHEMATICAL. LOGIC AND SET THEORY. REAL NUMBERS. Propositional logic. The language of propositional logic consists of. • a set of.

## Mathematical logic

Everything mathematicians do can be reduced to statements about sets, equality and membership which are basics of set theory. This course introduces these basic concepts. The course aims at familiarizing the students with cardinals, relations and fundamentals of propositional and predicate logics. Sets, subsets, operations with sets: union, intersection, difference, symmetric difference, Cartesian product and disjoint union. Functions: graph of a function.

The papers are listed in reverse chronological order, except that I put two surveys at the beginning to make them easier to find. PostScript or PDF. An expository talk, for a general mathematical audience, about cardinal characteristics of the continuum. Foreman, M. Magidor, and A.

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## 4 Comments

Alicia C.of a set. The second primitive notion of set theory is the notion of belonging. We three objects a, b, c, the set whose elements are precisely a, b, c is denoted by {a, b, c}. Designations are names used to refer to mathematical objects.

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