A sequence or series is a very fundamental idea and not easy to define in non-circular 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 well-known 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:

(a₁..a_n)=(b₁..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 i-th element of the one is the same as the i-th 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.