1.1 Sets and logic

There are essentially two ways in which we can think of a set S. Firstly, it can be regarded as a collection of mathematical objects $a,b,\dots$ , called constants, written

$$S=\{a,b,\dots \}.$$

The constants $a,b,...$ . may themselves be sets and, indeed, some formulations of set theory require them to be sets. Physicists in general prefer to avoid this formal nicety, and find it much more natural to allow for ‘atomic’ objects, as it is hard to think of quantities such as temperature or velocity as being ‘sets’. However, to think of sets as consisting of lists of objects is only suitable for finite or at most countably infinite sets. If we try putting the real numbers into a list we encounter the Cantor diagonalization problem – see Theorem 1.4 and Theorem 1.5 of Section 1.5.

The second approach to set theory is much more general in character. Let $P(x)$ be a logical proposition involving a variable x. Any such proposition symbolically defines a set

$$S=\{x\mid P(x)\},$$

which can be thought of as symbolically representing the collection of all x for which the proposition $P(x)$ is true. We will not attempt a full definition of the concept of logical proposition here – this is the business of formal logic and is only of peripheral interest to theoretical physicists – but some comments are in order. Essentially, logical propositions are statements made up from an alphabet of symbols, some of which are termed constants and some of which are called variables, together with logical connectives such as not, and, or and implies, to be manipulated according to rules of standard logic. Instead of “P implies Q” we write “if P then Q”, or $P\Rightarrow Q$. The statement “P if and only if Q”, or “P iff Q”, is written $P\iff Q$ and means

$$(P\Rightarrow Q)\text{and}(Q\Rightarrow P).$$

The quantifiers $\forall$ and $\exists$ mean “for all” and “there exists”. If $P(x)$ is a proposition, then $\forall xP(x)$ and $\exists xP(x)$ are propositions.

Mathematical theories such as set theory, group theory, etc. traditionally involve the introduction of some new symbols with which to generate further logical propositions. The theory must be complemented by a collection of logical propositions called axioms for the theory – statements that are taken to be automatically true in the theory. All other true statements should in principle follow by the rules of logic.

Set theory involves the introduction of the new phrase is a set and new symbols $\{\cdots \mid \dots \}$ and $\in$, defined by:

(Set1) If S is any constant or variable then ‘S is a set’ is a logical proposition.

(Set2) If $P(x)$ is a logical proposition involving a variable x then $\{x\mid P(x)\}$ is a set.

(Set3) If S is a set and a is any constant or variable then $a\in S$ is a logical proposition, for which we say a belongs to $S$ or a is a member of S, or simply a is in S. The negative of this proposition is denoted $a\notin S$ – said a is not in S.

These statements say nothing about whether the various propositions are true or false – they merely assert what are ‘grammatically correct’ propositions in set theory. They merely tell us how the new symbols and phrases are to be used in a grammatically correct fashion. The main axiom of set theory is: if $P(x)$ is any logical proposition depending on a variable x,

then for any constant or variable a

$$a\in \{x\mid P(x)\}\iff P(a).$$

Every mathematical theory uses the equality symbol to express the identity of mathematical objects in the theory. In some cases the concept of mathematical identity needs a separate definition. For example equality of sets $A=B$ is defined through the axiom of extensionality:

Two sets A and B are equal if and only if they contain the same members. Expressed symbolically,

$$A=B\iff \forall a(a\in A\iff a\in B).$$

A finite set $A=\{a_1,a_2,\dots ,a_n\}$ is equivalent to

$$A=\{x\mid (x=a_1)\text{ or }(x=a_2)\text{ or }\dots \text{ or }(x=a_n)\}.$$

A set consisting of just one element a is called a singleton and should be written as $\{a\}$ to distinguish it from the element a which belongs to it: $\{a\}=\{x\mid x=a\}$

As remarked above, sets can be members of other sets. A set whose elements are all sets themselves will often be called a collection or family of sets. Such collections are often denoted by script letters such as $\mathcal{A},\mathcal{U},$ , etc. Frequently a family of sets has its members indexed by another set I, called the indexing set, and is written

$$\mathcal{U}=\{U_i\mid i\in I\}.$$

For a finite family we usually take the indexing set to be the first n natural numbers, $I=\{1,2,\dots ,n\}$ . Strictly speaking, this set must also be given an axiomatic definition such as Peano’s axioms. We refer the interested reader to texts such as [4] for a discussion of these matters.

Although the finer details of logic have been omitted here, essentially all concepts of set theory can be constructed from these basics. The implication is that all of mathematics can be built out ofan alphabet for constants and variables, parentheses $(\dots )$ , logical connectives and quantifiers together with the rules of propositional logic, and the symbols $\{\cdots \mid \dots \}$ and $\in$. Since mathematical physics is an attempt to express physics in purely mathematical language, we have the somewhat astonishing implication that all of physics should also be reducible to these simple terms. Eugene Wigner has expressed wonderment at this idea in a famous paper entitled The unreasonable effectiveness of mathematics in the natura sciences [5].

The presentation of set theory given here should suffice for all practical purposes, but it is not without logical difficulties. The most famous is Russell’s paradox: consider the set of all sets which are not members of themselves. According to the above rules this set can be written $R=\{A\mid A\notin A\}$ . Is R a member of itself?

Proof

This question does not appear to have an answer. For, if $R\in R$ then by definition $R\notin R$ , which is a contradiction. On the other hand, if $R\notin R$ then it satisfies the criterion required for membership of $R;$ that is, $R\in R$

Avoiding the paradox

To avoid such vicious arguments, logicians have been forced to reformulate the axioms of set theory in a very careful way. The most frequently used system is the axiomatic scheme of Zermelo and Fraenkel – see, for example, [2] or the Appendix of [6]. We will adopt the ‘naive’ position and simply assume that the sets dealt with in this book do not exhibit the self-contradictions of Russell’s monster.

Referenced 1 time

• 1.7 Category theory(Cat1) A class whose elements are called objects. Note the use of the word ‘class’ rather than ‘set’…