We then took the chain to be a set of all things caused and called it the contingent set. — Miles
Ah, I didn't do that - I didn't call it 'contingent' (that would be the point at which the conflation occurs - the conflation I was warning against). For we then have these troublesome and distracting notions of contingency and necessity getting in the way (or being used in ways that will confuse).
If every event has a cause - and we are taking that to be self-evidently true - then because there cannot be an actual infinity of events, we must conclude that some events are not caused by events, but by substances.
We do not need to talk about contingency and necessity to get to this conclusion. I have reached it without invoking those concepts. All that is needed is the self-evident truth that all events have a cause, the self-evident truth that there are some events, and the self-evident truth that there cannot be an actual infinity of anything. Those claims are sufficient to get us to the conclusion that some events are substance-caused, rather than event-caused.
Whether any particular substance that does the causing exists of necessity, or does the causing of necessity, is really neither here nor there, so far as I can tell anyway. For I do not have to claim that all 'continent' events require a cause - just all events - or that all substances that cause things exist of necessity (just 'exist' is sufficient).
This we called the necessary existent by which we meant something non-contingent.
So far is the summary of what we said before. — Miles
Like I say, I do not think that's true. These terms 'necessary' and 'contingent' have just appeared at a certain point - I did not introduce them - and they seem to be referring not to actual necessity and contingency, but to other things. So, what you are calling 'contingent' are 'events'. But 'contingent' doesn't mean 'events' or 'set of all events' or whatever. And what you are calling 'necessary' are the substances that exist and are the ultimate causes of all events. But that is a strange use of the word 'necessary'. So again, I think what's happening is that completely different notions are being conflated.
If there are events, those events have causes. It doesn't matter whether the events are occurring contingently or of necessity - the same applies either way.
And there are no actual infinities of anything, so if there are events then ultimately their causal chains must trace to instances of substance causation, not event causation.
Again, no mention of contingency or necessity - for what I have just said is true regardless of whether the events are occurring contingently or of necessity.
And the substances that are doing the causing exist. That follows, but what does not follow is that they exist necessarily.
Meaning; given there is a world of events, we can conclude that in this world there necessary exists some non-contingent being. In other words given there is a world of events it is necessary that such an entity exists. — Miles
That, I think, does not follow. Here's what follows given my assumptions (those being 'every event has a cause' and 'there is no actual infinity of anything'). Given there are events, we can conclude that in this world there are substance causes. That is, there are substances and some of these substances cause events. Whether these substances exist of necessity or not is not something we can conclude from this argument. You are, it seems to me, committing the very mistake you highlighted earlier. You are going from 'given X, Y must exist" to "Y exists of necessity". Given that there are events, substance-causes must exist. But it does not follow from this that the substances in question exist necessarily.
So, in my view all of these things can be established:
There are substance-causes
The substance causes are simple
The substance causes are self-explanatory.
But I do not think any of that entails that they exist of necessity.