A system is a particular abstract theory defined by specifying certain basic concepts called its members and the axioms that relate them. The members are either elements, whose meanings are undefined, or they are constructs whose meanings can be defined within the system, in terms of other constructs and, ultimately, in terms of the elements.
This means that we could, if we so chose, eliminate a construct from all our statements, replacing it everywhere by its expression in terms of other concepts. The result would be to say the same things but in a more unwieldy fashion. A good reason for defining a new term is that it enables us to say neatly a lot of things that previously we could say only in a long-winded fashion. Unfortunately there is a tendency for unnecessary terms to get defined and to accumulate with obfuscating rather than clarifying effect.
A useful notation is that of the three dots ..., called ellipsis, which is short for ‘and so on’ or ‘et cetera’ and enables us to refer to a long sequence of symbols without listing them all. It is however implicitly assumed that any such statement is an abbreviation of an actual sentence that we could write out in full, so the sequence of missing items cannot be regarded as ‘going on forever’ whatever that might mean. [This is discussed further in the section on infinity.]
Systems S and T are said to be isomorphic if their members can be placed in correspondence (a to a', b to b', c to c', ...) in such a way that a statement A about members a, b, c, ... of S is true if and only if the corresponding statement A' about members a', b', c', ... is true in T. In other words, by changing the names of the components of the systems we can describe both systems in exactly the same terms—“The story is the same, only the names have been changed”.
A correspondence between systems that shows them to be isomorphic is called an isomorphism. An isomorphism can be regarded as a relation between the systems, or it can be regarded as an operation acting from one to the other of them, in either direction. Thus one system may be regarded as being produced from the other, i.e. as being a transformation of it, or one may be regarded as being used to describe the other, i.e. as being a representation of it. These are aspects of the same relationship, seen from different viewpoints.
Isomorphic systems are said to have the same structure. The structure of a system may be regarded as composed of all the systems isomorphic to it (the ‘constructionist’ approach), or as a property common to all the systems, or alternatively as represented by one example chosen as the standard (this seems to present the fewest logical difficulties). We denote the structure of S by |S|.
A set may be defined to be a constructless system. The concept of a set is fundamental to our ways of thinking, as is evidenced by the numerous alternative terms for the same concept: aggregate, class, collection, crowd, family, flock, group, herd, kind, range, selection, sort, type, and so on—some of which are restricted to use in particular contexts and some of which have special technical senses. The values of a variable constitute a set.
A notation sometimes useful is to abbreviate the expression ‘x is a member of the set S’ as: x∈S, using a special symbol based on the Greek epsilon to denote this relation of membership. To say that sets S and T are equal (written S = T) means the same as saying that any member of S is a member of T and vice versa. It is usual to denote the set with given members by enclosing the list of members within curly brackets, thus: S = {a, b, c, ..., k}. New concepts must always be defined in terms of concepts that already exist; under this assumption no system or set can be a member of itself. Failure to appreciate this can lead to paradoxes.
An important method of reasoning is that known in the Latin treatises on logic as reductio ad absurdum or in English reduction to absurdity. The procedure is to suppose that the proposition in question, A, is true, and then to deduce, on the basis of other already accepted propositions, a statement known to be false. This shows that the proposition A is incompatible with the already accepted assumptions, and is therefore also false.
Is there such a thing as a 'set of all sets'? Considering this question leads us to Russell's paradox. From any given things we can form the set with those things as its members. Suppose we can form a 'set of all sets', U. If so it would be a member of itself, since it is a set and therefore meets the membership requirement. Most sets we encounter however are not members of themselves (for instance a set of ants is not an ant). If we can form the set of all sets then we must be able to form the set of all sets that are not members of themselves, N. This would be a subset of U. If this set is a member of itself then it must be a set that is not a member of itself, since this is its membership criterion. On the other hand if it is not a member of itself it meets the membership criterion and must be a member of itself. Thus we have proved that it both is and is not a member of itself. We have generated a contradiction. It follows that our initial assumption, that there is a 'set of all sets', must be wrong. No such set exists. The paradox results because by defining a set, which is formed from 'given' (i.e. already existing) things, we are carrying out a process and thereby expanding the 'given' context and changing the meaning of 'all'.
The context within which a discussion takes place can be regarded as a
set called the full set I. Within a given context there is also a uniquely
defined null set O, that has no concepts from that context as members.
It may be objected that a set, by definition, must have members, but
although the concept of a null set seems paradoxical there is in fact no contradiction
here, it is just a matter of convention. The apparent paradox results from the use of
mutually incompatible conventions in normal and mathematical usage of English.
The full set of elements of a system form a set which may be regarded as the space within which the system is constructed. The size of this space is an important feature of the structure of the system.
Isomorphism between sets is sometimes called equipollence. The structure of a set is its size (or cardinality or the number of its elements). Thus, in terms of the structure theory of mathematics, numbers describe the structure of systems of discrete objects (sets). Two sets have the same structure if their members can be paired off with one another (in other words the sets can be placed in one-to-one correspondence).
A family is a set of sets, that is a set whose members are themselves sets. In a separate section on 'Set Algebra' we define relations and operations between sets, and examine their properties. They provide examples of Boolean Algebras.