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.