Sequence : Several things that
all fall under a relation that is assymmetric and
transitive.
A sequence or series is a very fundamental idea and not easy to define
in noncircular terms. Here the circularity, or at least its appearance,
resides in the relation, for these tend
to be involved in terms of sequences or tuples.
Sequences are intuitively quite wellknown to human beings, since so
much of what they know comes in sequences. Three examples are the
sequences that form the stages of age, the sequences of letters that are
words and the sequences of words that are statements.
A law for sequences can be formulated as follows:
(a_{1}..a_{n})=(b_{1}..b_{m}) IFF n=m &
(a)(b)(i)(a_{i}=b_{i})
Any two sequences are equal iff they have the same number of elements
and each ith element of the one is the same as the ith element of the
other. Put otherwise, identical sequences have the same elements in the
same order.
A relation R that is assymmetric and transitive has the
following properties in terms of what it implies for any of its
elements:
(1) R is assymmetric: For any x and y, if Rxy, then not Ryx
(2) R is transitive: For any x and y and z, if Rxy and Ryz, then Rxz
The notions 'smaller than' and 'greater than' from standard
arithmetics or algebra have these properties.
The point of introducing such a relation, which is called an
ordering relation, is that in terms of it all the elements of a
sequence come before and after other elements of the sequence, except
the first and the last, which have the respective defining properties of
coming after none of the other elements of the sequence and before all
others of it and of coming coming before none of the other elements of
the sequence and after all others of it.
