Unordered or ordered

1. I'm commenting from this article by James D. Fearon of Stanford University

2. I'm curious to know how we can define a set as an unordered collection then we can talk about "ordered sets." If a set is defined as unordered there cannot be ordered sets.

3. I know that, I'm sure that, there is an explanation, in fact I suspect that there exists a "Zermalo's axiom" that lets us write ordered sets even though we defined sets as unordered. Or we may say "ordered sets are not really ordered in the sense of the original definition so there is no contradiction."

4. So I'd like to know how mathematicians rationalize ordered sets even though an ordered collection in mathematics is called a sequence.

5. Let's look at an explicit example.

6. On page 1, Fearon writes:

The order of elements in the set does not matter.

7. So the set $\mathbf{A} = \{\; 0, 1, 2, 3\;\}$ and $\mathbf{A} = \{\; 1, 2, 0, 3\;\}$ are the same set.

8. A set is defined as an unordered collection of elements.

9. So far so good. A set is unordered. Order of elements does not matter. But now let's look at the next page where he explains infinite sets.

10. He first gives an example with the set construction notation:

$$\mathbf{S}= \{\; s : 0 < s < 1\; \}$$

11.  So, $\mathbf{S}$ is a set containing elements $s_n$ between $0$ and $1$.

12. Fearon explains:

In this method a set is defined by specifying the property or properties that characterizes or is true of all elements of the set.

13. This is standard mathematics too.

14. Then, he gives another example:

Occasionally, you will also see notation like the following for an infinite set:

$$\mathbf{T} = \{\; 0, 1, 2, 3\; \ldots \;\}$$

where it is understood that the ellipsis means that the set proceeds as indicated.

15. But $\mathbf{T}$ is an ordered collection, so it cannot be a set. To call $\mathbf{T}$ a set contradicts the standard definition of a set.

16. $\mathbf{T}$ is not a set because it is ordered. Ordered collections are called "sequences". So why are mathematicians calling a sequence a set?

17. If $\mathbf{T}$ were to be a set I could write it like

$$\mathbf{T} = \{\; 3, 0, 2, 1\; \ldots \;\}$$

18. But now ellipsis makes no sense. We cannot say "the ellipsis means that the set proceeds as indicated."

19. $\mathbf{T} = \{\; 3, 0, 2, 1\; \ldots \;\}$ has no order so we cannot know how to continue this set. Ellipsis implies and assumes order, when there is no order ellipsis makes no sense.

20. So, which one of Zermalo's famous axioms allows us to write ordered sets even though we defined a set to be an unordered collection?

21. I can almost hear the answer. "An ordered set is not really ordered, it is Zermalo-ordered, or Dedekind-ordered or Cantor-ordered which is according to $N$'th order logic is not really order at all. Where is this nasty odor of scholastic sophistry is coming from?