Meaning of 'set of well-ordered sequences'

Question

I'm trying to make sense of a construction of a module given in the following research paper: A New Construction of the Injective Hull, Fleischer, 1968.

On the second page, a module $F$ is constructed, with the definition of the underlying set being 'the set of those well-ordered sequences $\{f_\alpha\}$ of ...', which seems unclear to me.

What is a 'well-ordered sequence'? My understanding of a sequence is that it is a function with domain $\mathbb{N}$. Of course $\mathbb{N}$ is well-ordered, but there is no need to refer to a sequence as well-ordered. So I guess this is referring to something slightly different, but I do not see any well-ordering going on anywhere so am at a loss. What might this be referring to?

Answer 1

I presume a well-ordered sequence in $F$ means a map $\alpha \to F; a \mapsto f_a$ for some well-ordered set $\alpha$.

For $\alpha = \mathbb N$ we get a normal sequence. For $\alpha$ finite we get a finite set of elements of $F$. For $\alpha = \omega_1$ we get an $\omega_1$-sequence which is like a normal sequence but longer.

The set of all well-ordered sequences includes all of the above.

There are cardinality problems with defining such a set, since you quickly end up considering the set of all sets. But for any given application it is often fine to only consider all well-ordered sets below some given cardinality depending on the size of the other objects under consideration.

Answer 2

In this context, a well ordered sequence is just a sequence $(f_\beta)_{\beta < \gamma}$ where $\gamma$ is a well ordered set (and hence $\gamma$ is isomorphic to an ordinal so wlog you can assume that $\gamma$ is an ordinal).

NB:

• $\mathbb{N}$, a.k.a $\omega$, is a well ordered set, hence every sequence indexed by $\mathbb{N}$ is indeed well ordered.
• Every finite sequence is well ordered.