Definition of a sequence.

Question

I have seen many books defining sequence as a function whose domain is N, the set of all natural numbers.

I really don't appreciate such 'less-general' definition.

Consider this: A sequence is an enumerated collection of objects.

My Observations: 1. Enumeration of a collection implies the collection is indeed well defined. For otherwise: we really don't know what objects are qualifying to be its members, let alone their enumeration!

2. It includes the case of a finite sequence as well! Even the empty sequence can be defined now.

3.By definition of collection, repetitions of the elements are allowed.

Is this definition fine? Please suggest the mistakes.

The main motivation to do all this is to not see a sequence as a function. Rather it in itself is something different structure like set, group, e.t.c. By enumeration I mean listing all elements of a collection from left to right, one by one. Eg. The two sequences (1,2,3,...) and (2,1,3,4,5,...) are different because their elements are enumerated (arranged) in different ways. By countable collection I mean a denumerable or finite collection. However one can arrive to the usual definition from this. If there is something wrong in my definition, please help me to see that this does imply something absurd.

Answer 1

Your proposed definition is based upon the concept of enumeration. Can you define enumeration in a way which is essencialy different from the definition of sequence? If not, then what's the problem with the usual definition of sequence? Otherwise, tell us what your definition is.

Answer 2

Your definition is very problematic because the term "enumerated" is vague and you have not defined it. The entire point of giving a definition like "function whose domain is $\mathbb{N}$" is to have a perfectly precise definition, in terms of other concepts that already have precise definitions.

Here is one way to salvage this definition which is pretty standard. Define a sequence to be a function whose domain is either $\mathbb{N}$ or the set $\{m\in\mathbb{N}:m<n\}$ for some $n\in\mathbb{N}$. Or more concisely, a sequence is a function whose domain is an initial segment of $\mathbb{N}$ (an initial segment being a subset $S$ of $\mathbb{N}$ such that $s\in S$ implies $n\in S$ for all $n\in\mathbb{N}$ such that $n\leq s$).