Hey, you got it! (I'm not the least bit surprised; you seem to be quite a bit more intelligent than I am; at the very least, you know quite a bit more about mathematics, set theory, and so on, stuff I'm hazy on at best).
I think that it's evident that I badly phrased my arguments. My sincere apologies for that. I am, as yet, a beginning student of informal logic, and I trip over myself occasionally trying to make difficult what should be simple (and vice-versa).
You said:
Rephrasing in terms of set theory:
P3: The set of all natural things is not empty. [Identical to Premise 1 above]
P4: The set of natural things that has always existed is empty.
C2: Every natural thing that exists now began to exist at some point in time.
I accept this as a valid re-wording of my second argument. As you've written it, it states what I was TRYING to say from the beginning. Your conclusion equals mine: That either there is some natural thing that has always existed or there is no natural thing that has always existed.
Moving on:
Argument.
The second and fourth premise are the antithesis of each other; it takes the form of ‘A’ or ‘Not A’ (A ∨ ¬A). There is no false dichotomy here; either there has
always been at least one natural thing x, or there has not. These premises stand on their own, and cannot be reduced
ad absurdum.
Both P1…C1 and P3…C2 are logically valid arguments in the way in which they are constructed. The problem with Premise 1 is that we have neither empirical evidence nor repeatable tests for
any natural thing ‘x’ which has
always existed.
If a person’s position is that, “I don’t have enough information to decide whether it’s C1 or C2; therefore I am skeptical of
both,� under the circumstances presented here, this is a form of dishonest (or unreasonable) skepticism; there will
never be more information concerning C1.
If a person’s position is that, “I’m skeptical of C2 because there
might be some natural thing out there which has
always existed (i.e., some infinite thing),� then their position is an argument
ad ignorantum. Certainly such a thing
might exist; but in the same vein, pink unicorns or leprechauns
might exist. Just because it is
possible for something to exist is not reason enough to be skeptical of a statement denying such.
Since the first conclusion
cannot be accepted under the principles established here – that for a thing to be known and/or accepted, there must be empirical evidence of it and/or repeatable tests of it – that leaves the second conclusion. The subject at hand, the answer is either C1 or C2.
Since C1 has been eliminated, by virtue of the process of elimination and logic, the answer is therefore C2, that some natural thing ‘x’ {quarks, energy, particles, etc.} has
not always existed (the set of natural things that has always existed is empty), i.e., that
every thing 'x' which exists began to exist at some point in time. This concludes the first section of my argument.