The Principle of Contingent Causation: The Impossibility of Infinite Regress.

[Angrboda]

Ok, so in Good Thomistic Fashion (as in the Summa), let's address the objections first:

The first one, quite common in this thread, seems to be something along these lines.

Objection 1: "The First Cause is not yet proven to be Jesus Christ, therefore I don't have to admit it."
[Alternately, "Since Saint Thomas believed the First Cause is Jesus Christ, ergo ... etc" still fallacious].

Response: that's a Non Sequitur. One could come to Jesus Christ in say premise 10 or so. That does nothing to show step 3 or 4 of the Argument is not logically valid and follows necessarily from the preceding premises. The only way for Atheists to show that is to contest one of those same steps.

Steps 1 to 4 as outlined in the OP could be affirmed by people from various persuasions, including Deists, Hindus, some Buddhists, Muslims, and Christians, beside others. It says nothing, as yet, about the Personality, or Goodness, of the First Being/First Cause, but only about Its Existence.

The Moral Argument, from Conscience, would help in establishing the Creator's Goodness, and thus His Personality as well; this argument is focused first on establishing that a First Cause of the Universe exists, while the Universe itself, and all things within it, exist contingently and not necessarily.

For a secular example: if one proved Pythagoras' Theorem by appealing to Pythagoras' Theorem, that would be circular. But not if one proved it from first principles that can be independently known. And the fact that one may later build on PT, or apply it in various ways, does not diminish the proof.

An argument not given need not be refuted. If you're depending upon another result, you need to introduce it. There are three problems with this. First, your proof from morality was not successful because it was an argument basically that you "feel" there are morals and therefore there are; that's not valid. Second, you cannot assume that the cause of morals and the first cause are one and the same simply because you want it to be so. Nothing in your argument from morals shows that the entity responsible for morals is a first cause, and nothing in your argument for a first cause shows that cause is responsible for morals. Assuming they are one and the same without argument is invalid. And finally, even if you had shown that, a first cause that is responsible for morals still could be something other than a god. Again, an argument not in evidence need not be answered. You and Bel talk about other traits of your god being shown in other arguments. For some reason they never end up in the same argument, and the reason is simple, because the logical disconnects would be obvious if they were. Asserting that they can be connected up without actually showing the connection is just an ipse dixit argument.

Objection 2: "Even if a Necessary First Cause of all things exist, this First Cause could allegedly be the universe/something else natural itself."

This objection does not take into account what we proved about B1, the First Being. The First Being is non-contingent. Contingency is shown in 2 ways. First, (1) something that is contingent began to exist; (2) second, something that could even conceivably cease to exist is certainly contingent. Now, most Scientists agree the Universe began to exist, in the Big Bang, and also, almost all agree that, at least in theory, that the Universe either will or could cease to exist one day. Therefore, (3) the Universe exists contingently, not necessarily. Why is this relevant? Because we proved in the OP, mathematically, that B1 exists Non-Contingently. Therefore, B1 cannot be the Universe, but refers to the First Cause of the Universe itself. 

Recall the proof of the non-contingency of B1. Since there is no B0, and Bn by definition is contingent only if it is dependent on a prior being Bn-1, it follows that the First Being in existence is non-contingent; the First Being exists necessarily. And for the above reasons, that the visible universe does not demonstrate the necessary properties of B1, the First Cause is not the universe; rather, it is the First Cause of the Universe that exists necessarily or non-contingently.

I'm not sure that being able to conceive of something not existing is a defining feature of contingent things. I think most of us atheists can conceive of your god not existing. By that measure, your god is contingent. No, being able to conceive something is insufficient because we can conceive of things that are not in fact possible. So while you might be able to conceive of the universe not existing, that does not mean the universe can cease to exist. Otherwise your god fails the same test. And you still haven't shown that anything in this universe is in fact contingent. The atom in my finger may have once been an atom in a cheeseburger which was once an atom in a cow which was once an atom in a star and so on. Like the mistake most theists make with the Kalam, you mistake change in form with change in ontological existence. These "beings" you imagine as being in series aren't in series at all necessarily, they could all simply be different forms of one non-contingent thing, the universe.

Objection 3: I, an Atheist, affirm, just like that, that an Infinite Number of Beings can exist (and the proof for this is?). Therefore, even if the argument holds for all finite n, it fails when n is infinite. Therefore, I don't have to believe in a First Cause".

Once more, notice how Atheists exempt themselves from the obligation to prove what they claim (in this case, the absurdity that an Infinite Number of Beings can exist) and the necessity of demonstrating that in a series of logical steps. Then, they demand that we Theists should do so.

We Theists can, but they cannot. Anyway let's examine this objection.

Firstly, notice we haven't assumed anything about N. N indeed holds good for all values, whether it be 2 or 100 trillion. That shows any and all contingent beings in existence, whether today or 4 thousand years ago, should have been able to use these steps to come to the knowledge of the First Cause. What of the claim that the number of beings in existence could be actually infinite?

I mentioned this objection in the OP and answered it briefly thus: "The only alternative to the existence of an actual first being is an infinite series of contingent beings, but that is impossible because an infinite series never ends; and if there were an actual infinite of real beings, we would never have gotten to the present moment; again, an infinite series cannot be formed by successive addition, because no matter how [many] beings you add to each other, whether it is 1 or 1 trillion, n will always be finite. Therefore, granted that we got here, granted that we are 1 in a series of contingent beings, the number of beings in existence is finite."

Your argument here is rather pedestrian. Actual experts on the subject do not agree with you. What credential do you have to make such assertions? Like many, including William Lane Craig, you confuse properties of finite sets with those of infinite sets and you cannot reason from one to the other. What is true of a finite collection is not necessarily the case with an infinite collection. All the examples you give are examples of critiquing the possible existence of infinite sets by showing that an operation on a finite set yields an absurdity or attempts to bridge the gap from a finite exemplar to an infinite collection.

Moreover, you simply misunderstand the burden of proof. The atheist is not needing to demonstrate that a specific counterexample is actually the case, only that it is possible. You are making an affirmative case. The possible existence of a counter-example is fatal to your argument. That is all the person arguing against you need show. Either you actually don't understand the burden of proof here, or you do and you are illicitly trying to shift the burden of proof. The latter would be an argument in bad faith.

Again, a simpler refutation is from Modern Science; since most Physicists are agreed the Universe is of Finite Age (roughly 13.7 BN years), it clearly follows that an Infinite Number of Beings could not have existed in said finite time. That such arguments should be seriously proposed by Atheists, in order to undermine Premise 2, shows how weak the Atheistic position ultimately is. 

You betray the fact that you don't know what you are talking about. No, most physicists do not agree with this. Physicists who understand the subject agree it is simply unknown.