There are several terms in the common language that are associated with the concept of
order. Here, I mention those that are not capable of being scientifically defined. Attempts to do so invariably exhibit circularity. The descriptions made in the attempted definitions employ synonyms. This makes the descriptions circular. There is a simple reason for this: These meanings are too basic to be described by more basic terms.
The concept of order is inherent in matters of
time,
distance,
rank, and other common terms. One need not mention the term
order when using terms such as
before,
after,
left,
right,
next in line or
next in time. The order of time is thoroughly conspicuous.
The term
next can mean that something is juxtaposed to something, or it can have the meaning associated with order in which it means
next in order. This is the sense of the term that will be used here, unless explicitly stated otherwise.
Like these terms that are associated with order, the term order itself is not capable of being scientifically defined. It is learned through experience in the world.
To say that the applause event is next in order to the curtain falling event is unambiguous.
Please keep in mind that elements can be events, actions, statements, or things of any kind. Also, lacking an axiom of infinity, you can call sets here finite.
Definition - S is an ordered set if and only if the following statements are true:
S is a set containing at least a pair of elements (at least two elements).
A single element j in S is the only element in S that is not next in order to some other element in S.
A single element k in S is the only element in S such that there is no element in S which is next in order to k in S.
If element y is next in order to element x in S, then no element other than y is next in order to x in S.
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Notation - In the case of an ordered set, the curly brackets that are normally used to enclose elements in a set are replaced by parentheses. Unless otherwise specifically required, if y is next in order to x, then y is written to the right of x. Here is an example:
S = (t, u, v, w, x, y, z)
In 1921, Kazimierz Kuratowski offered the now-conventional definition of the ordered pair (a, b):
(a, b) = {{a}, {a, b}}
This equation cannot be used in the system of definitions because the axioms of traditional set theory that give this equation meaning are missing.
Jim Adrian
