We can use whatever words we like to define things, but when you use a term with your own definition that doesn’t conform to what everyone else in the field means by it, it’s going to cause confusion.
I agree it might lead to confusion. So, we need to define our terms. I did labor to define and clarify what I meant by “essentially subordinated series.” Provided definitions and clarifications are made, I don’t think we should hesitate to use terms in a different way from their common acceptation, so that we can advance our knowledge. Had Einstein hesitated to use familiar words in a different context, we will not know about his Theory of Relativity, for he spoke of “space” and “time” as relative rather than absolute, which was not the common understanding of his peers at the time.
Suffice it to say, much of what you’re describing as essentially ordered is what St. Thomas and many Thomists hold can proceed to infinity without issue,
I agree. An example of this is the Father and Son example, which St. Thomas and most Thomists consider as an accidentally ordered series that can proceed to infinity. But that is only because they considered this series as an accidentally ordered series
in respect of generation. They did not realize that it is also an essentially ordered series
in respect of being. If a man, A, begets a son, B, who then begets a son C, then I completely agree that C depends on A only accidentally in respect of generation, because B generates C, not as a son of A, but as a man. However, I can also say without hesitation, that C depends on A essentially
in respect of being. For, C is a man (and not a lizard), because his father, B, is a man; and B is a man (and not a lizard) because his father, A, is a man. Therefore, although C only depends accidentally on A for being born or generated, C depends essentially on A for being a man. You cannot separate the generation of an individual from its being, because every organism is not generated indeterminately, but is generated as a particular kind of being.
So, if we consider a chicken that comes from an egg, which comes from another chicken that comes from another egg, etc. (a variation of the father and son example), my contention is that there was a first chicken or first egg (but I will let science determine that). I believe this to be the case because I believe that the world was created, and that it was neither infinite nor everlasting. But that things have a beginning in time is
an article of faith, not a proposition demonstrated by metaphysics, as St. Thomas said. So, if the chicken and egg example is carried to infinity, it still won’t bother me. Since the dependence of each member of the series to the previous member is
essential in respect of being, the entire series cannot exist without an Uncaused Cause of Being. Therefore, if my definition of an essentially ordered series is used in any of the Ways of St. Thomas, it will always work. The arguments of St. Thomas are not jeopardized just because I have a different definition of an essentially ordered series.