by **Magnus Anderson** » Tue Jan 28, 2020 1:25 am

It has been claimed that it's a contradiction in terms to say that an infinite sequence has a beginning and an end.

An infinite sequence has no end, so you cannot say that it has an end.

Well, you actually can, provided that the first occurrence of the word "end" and the second occurrence of the word "end" mean two different things (i.e. provided that they refer to two different ends.)

Don't be fooled by homonyms.

\(S = (e_1, e_2, e_3, \dotso, e_L)\) is one such sequence. It's an infinite sequence with a beginning and an end.

Note that a sequence with no repetitions is no more than a relation between the set of positions and the set of elements.

\(S = (e_1, e_2, e_3, \dotso, e_L)\) is a relation between the set of positions \(P = \{1, 2, 3, \dotso, \infty\}\) and the set of elements \(E = \{e_1, e_2, e_3, \dotso, e_L\}\).

Note that sets have no order. This means that, when visually representing a set, you can place its elements anywhere you want. This means that \(P = \{1, 2, 3, \dotso, \infty\} = \{\infty, 1, 2, 3, \dotso\}\). The same applies to \(E\). By moving the last element of the set to the beginning of the set, there are no longer any elements after the ellipsis, so there is less to complain about (:

Let's represent the sequence as a set of pairs \((\text{position}, \text{element})\). \(S = (e_1, e_2, e_3, \dotso, e_L) = \{(\infty, e_L), (1, e_1), (2, e_2), (3, e_3), \dotso\}\). And voila! There is nothing beyond the ellipsis anymore, so absolutely nothing to complain about (((:

"The last position in the sequence" refers to the largest number in the set of positions \(P\). Either there is such a number or there is not. In the case of our sequence, there is such a number and it is \(\infty\).

This is not the same as "The number of elements in the set of positions \(P\)". This is an entirely different thing. In the case of our sequence, the number of positions is infinite (i.e. there is no end to the number of elements.) It's also not the same as "The last element in the set of positions \(P\)". No such thing exists, not because the set is infinite, but because sets have no order.