Introduction to Mathematical Logic

Extended Edition

Micha?? Walicki

• Verlag: WSPC
• Jahr: 2016
• ISBN:
  9789814719988
• Thema: Mathematik
• Thema: Logik in der Mathematik

This is a systematic and well-paced introduction to mathematical logic. Excellent as a course text, the book presupposes only elementary background and can be used also for self-study by more ambitious students.

Starting with the basics of set th

Part I

ELEMENTS OF SET THEORY

Chapter 1

SETS, FUNCTIONS, RELATIONS

1.1SETS AND FUNCTIONS

a background story

A set is an arbitrary collection of arbitrary objects, called its members. One should take these two occurrences of “arbitrary” seriously. Firstly, sets may be finite, e.g., the set C of cars on the parking lot outside the building, or infinite, e.g. the set N of numbers greater than 5.

Secondly, any objects can be members of sets. We can talk about sets of cars, blood-cells, numbers, Roman emperors, etc. We can also talk about the set X whose elements are: my car, your mother and number 6. (Not that such a set is necessarily useful for any purpose, but it is possible to collect these various elements into one set.) In particular sets themselves can be members of other sets. We can, for instance, form the set whose elements are: your favorite pen, your two best friends and the set N. This set will have 4 elements, even though the set N itself is infinite.

A set with only one element is called a singleton, e.g., the set containing only planet Earth. There is one special and very important set – the empty set – which has no members. If it seems startling, you may think of the set of all square circles or all numbers x such that x < x. This set is mainly a mathematical convenience – defining a set by describing the properties of its members in an involved way, we may not know from the very beginning what its members are. Eventually, we may find that no such objects exist, that is, that we defined an empty set. It also makes many formulations simpler since, without the assumption of its existence, one would often had to take special precautions for the case a set happened to contain no elements.

It may be legitimate to speak about a definite set even if we do not know exactly its members. The set of people born in 1964 may be hard to determine exactly but it is a well defined object because, at least in principle, we can determine membership of any object in this set. Similarly, we will say that the set R of red objects is well defined even if we certainly do not know all its members. But confronted with a new object, we can determine if it belongs to R or not (assuming, that we do not dispute the meaning of the word “red”).

There are four basic means of specifying a set.

(1)If a set is finite and small, we may list all its elements, e.g., S = {1, 2, 3, 4} is a set with four elements.

(2)A set can be specified by determining a property which makes objects qualify as its elements. The set R of red objects is specified in this way. The set S can be described as ‘the set of natural numbers greater than 0 and less than 5’.

(3)A set may be obtained from other sets. For instance, given the set S and the set S′ = {3, 4, 5, 6} we can form a new set S″ = {3, 4} which is the intersection of S and S′. Given the sets of odd {1, 3, 5, 7, 9, …} and even numbers {0, 2, 4, 6, 8, …} we can form a new set

by taking their union.

(4)Finally, a set can be also given by means of the rules for genera...