Smith san francisco state university these notes outline some set theory on which many parts of mathematics are based. The set containing no elements is known as the empty set. Aside generic interpretations, set theory has a standard kind of interpretations into itself, where each set is interpreted by the class meta set of its elements the synonymous set and meta set, i. Mobile device interfaces the mobile interface is the ultimate recording companion for the musician on the move. Halmos was quite painful to work through, because there was little mathematical notation. I worked my way through halmos naive set theory, and did about of robert vaughts book. Suitable for all introductory mathematics undergraduates, notes on logic and set theory covers the basic concepts of logic. The following notation is used to show set membership means that x is a member of the set a means that x is not a member of the set a. It is thus a metafunction from a nite metaset, to the universe. Implementation of bourbakis elements of mathematics in coq. Endertons elements of set theory and with the central results of intermediate logic completeness, compactness, lowenheimskolem, nonstandard models, godels incompleteness theorems will be presupposed.
A little set theory never hurt anybody matthew saltzman department of mathematical sciences clemson university draft. The purpose of this module is to introduce language for talking about sets, and some. Set theory \a set is a many that allows itself to be thought of as a one. In this paper, a systematic and critical study of the fundamentals of soft set theory, which include operations on soft sets and their properties, soft set relation and function, matrix representation of soft set among others, is. Usually we denote sets with uppercase letters, elements with lowercase letters. Cantors set theory was controversial at the start, but later became largely accepted.
S in etcs has no properties except that it is an element of s and is distinct from any other elements of s. Thenatural number structure in etcs is a triad of a set n, a selected. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. Set theory notes on h endertons elements of set theory i. It is used as a foundation for many subfields of mathematics. They are not guaranteed to be comprehensive of the material covered in the course. Set theory if we are interested in elements of a set a that are not contained in a set b, we can write this set as a. Fundamental to set theory is the notion of sets have members, also. Elements the objects in a set are called elements let s emily, kimerah, katherine emily is said to be an element of set s because she is part of that set the shorthand notation for this is emily. Sets the notions object, set, and membership are used in this theory without definition. For those of you new to abstract mathematics elementary does not mean simple though much of the material.
Then by the axiom schema of comprehension, there is a set bdf x2vw g. B for example, if s is the set of all juices in the supermarket, and t. What appeals to me most about descriptive set theory is that to study it you must reallyunderstandso many things. The author then develops the notions of transfinite induction and descriptive set theory, with applications to the theory of real functions. Indeed, one way to state the axioms of probability involves set theory. Cantors first proof that infinite sets can have different. An elementary theory of the category of sets 3 the sets of etcs are abstract structures in exactly this sense. Set theory for the working mathematician london mathematical. The usual set theory of zermelo and fraenkel is not entirely neglected. With the introduction to set theory fundamentals in the previous article, we are all set to explore the advanced realms of set theory through maxima. A study of the fundamentals of soft set theory onyeozili, i.
If a set contains an ordinal, then it contains a minimal ordinal. Tuples, families a tuple or ntuple, for any integer n is an interpretation of a list of nvariables. Enderton elements of set theory, academic press, 1977. Notes on logic and set theory cambridge mathematical. A set is any unordered collection of distinct objects.
S empty set in virtue of the principle of extensionality, there can be at most one set with no members. Projectteam marelle research report n 6999 version 5 initial version july 2009 revised version december 2011 225 pages abstract. Philosophy of set theory university of california, irvine. Martins axiom, the diamond principle, and elements of forcing. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. August 19, 2019 1 introduction the fundamental ideas of set theory and the algebra of sets are probably the most important concepts across all areas of mathematics other than the algebra of real numbers.
The second primitive notion of set theory is the notion of belonging. Hardegree, set theory an overview 1 of 34 set theory an. The formal axiomatic system of ordinary set theory zfc. Specifically, set theory postulates the existence of a set with no members. In particular, there have been objections to its use of infinite sets. This alone assures the subject of a place prominent in human culture. Set theory 1 set theory one of the main objectives of a statistician is to draw conclusions about a population of objects by conducting an experiment. Basic concepts of set theory, functions and relations. The empty set we said earlier that sets have members, but there is an exception.
Aside generic interpretations, set theory has a standard kind of interpretations into itself, where each set is interpreted by the class metaset of its elements the synonymous set and metaset, i. View notes endertonsolutions from math 361 at rutgers university. To express the relation of membership, we use a stylized epsilon symbol. Implementation of bourbakis elements of mathematics in. For instance, the set of all planets in the solar system, the set of all even integers, the set of all polynomials with real coe cients, and so on. For a property p and an element sof a set s, we write ps to indicate that shas the property p. The notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is, but we can give an informal description, describe. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. I later discovered endertons elements of set theory and i rec.
We have already worked out the basic set creation techniques and some basic set operations provided by maxima. A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Basic set theory a set is a many that allows itself to be thought of as a one. The main connection to mathematical set theory is the use of the vocabulary of set theory to talk about finite sets. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. The purpose of set theory is not practical application in the same way that, for example, fourier analysis has practical applications. There is a study of somewhat more advanced topics in. This set is for the test over set theory on december 8th, 2011.
In the past few years, the fundamentals of soft set theory have been studied by various researchers. But even more, set theory is the milieu in which mathematics takes place today. Georg cantor in the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains in predicate logic form sets as well. We believe that it is possible to put the whole work of bourbaki into a computer. A 2tuple is called an oriented pair, a 3tuple is a triple, a 4tuple is a quadrupletuples of a. Psgindicates that the set aconsists of all elements sof shaving the. In the areas pertaining to statistics, it is particularly used in probability. The notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is. When you do the symbol problems, draw the symbol down somewhere and type in for the answer 3 periods. The strength of mac lane set theory welcome to dpmms.
This book is dedicated to prabir roy, louis mcauley, jonathan. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. X meaning x belongs to the set x, or x is an element of x tipically we use capital letters to designate sets and small letters to designate elements of a set. Goldrei classic set theory, chapman and hall 1996, or h. The rest will be sketched in readings andor in class. Musical set theory is best regarded as a field that is not so much related to mathematical set theory, as an application of combinatorics to music theory with its own vocabulary. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. The default requirement for those taking the course for a grade other. X for some ordinal then the set z of elements of x which are ordinals is nonempty, and applying the axiom of foundation to this set we obtain an ordinal. There is a study of somewhat more advanced topics in set theory at the. Try one of the apps below to open or edit this item. The final part of the book presents the tools of modern set theory. B for example, if s is the set of all juices in the supermarket, and t is the set of all. These objects are called the elements or members of the set.
1117 1170 888 456 1041 104 190 686 904 83 534 1440 524 622 1502 1144 28 1395 1403 1371 1163 1086 838 874 509 1012 976 1155 1385