The problem you have is that your statement “…there cannot possibly be an explanation for [the universe” is simply a statement of opinion, not a statement of logic.Until you explain, in logical terms, how a contingent thing or set cannot have an ultimate necessary cause, this is just your opinion, no more. Indeed, it is your opinion against formal logic’s forms.

But anyway, (I think) we both agree that “the universe” the set of things which do, in fact, exist. And we agree the universe is contingent: there’s no logical, a priori reason that the universe has to be the way it is, or that anything has to exist at all. The point I’ve been trying to make is that, although the universe is contingent, there cannot possibly be an explanation for it, due to the very definition of “the universe”. It will never be possible to give an answer (natural or otherwise) to the question “why is there something rather than nothing?”.

We can only answer “why are things the way they are?”-type questions locally, not globally for the universe as a whole.

As for the Axiom of Choice: I was a bit confused about this issue. I believe the problem is that I had been thinking of the causal chain model I had laid out as a toy model of causality, with time divided into discrete chunks, and you had thought I meant it as a full, proper model reflecting the continuous nature of time. It’s true that if time is continuous, you can’t point identify a single point in time as the cause of the next point in time, since there are always more points between them. So, it’s necessary to slightly amend one’s view of what causality is. Basically, you want to define a time-evolution operator, U(X(t),T), which takes the current state of the world, X(t), and deterministically evolves it by an amount of time T (which may be negative or positive) to the state X(t+T). In this picture, you can sensibly say that the state X(t) is caused provided there exists some negative number, -tau, so that the action of U on X(t) is defined for all T in the interval [-tau,0]. (That, is if you can’t evolve the state X backwards in time at all, then it is not caused.) None of this seems terribly relevant to the Cosmological argument, which is why I initially gave a simple toy model of causality.

Please don’t confuse ‘existence’ and ‘contingency/necessity’; You are shifting from defining the universe as ‘everything that is contingent’ (which is the definition in the argument) to ‘everything that exists’ (your statement). These are two different definitions! *Of course* the argument is meaningless if you alter definitions!

Your argument “But the fact that the universe is uncaused does not preclude all the individual events within the universe from being caused” is very similar to Bertrand Russell’s ‘just because individual brinks are small doesn’t mean that the wall is small’. The reply was ‘we aren’t saying the wall is small, we are saying the wall is *brick*”.

In the end, the Cosmological argument succeeds in a very important way – if you accept certain assumptions, there is a logical proof of the existence of an uncaused entity that exists ‘outside’ the contingent/observable universe. And there is no method of proving those assumptions untrue. Thus, belief in God, so defined, is *just as logical* as unbelief. In short, you aren’t forced to accept the existence of God, but you *are* forced to accept that those who do accept the existence of God are not acting in an illogical or irrational manner.

By its very nature, the universe cannot have a cause or explanation for its existence. Why is this the case? What we mean by “the universe” is essentially “the collection of everything that exists”. Suppose that the universe was caused by some thing X. This thing X must be a real existing thing, but must lie outside of the universe in order to be its cause. This a contradiction: by definition, nothing lies outside the universe. (Yes, this argument is very similar to what Williams said; this is what I meant by when I mentioned that Williams inadvertently shows that it is misguided to look for a cause for the universe. But the fact that the universe is uncaused does not preclude all the individual events within the universe from being caused.)

Sure, you can use a modified definition of “universe” which limits its scope and allows some things to exist outside it. But then the “meta-universe” (consisting of the original universe together with these “outside things”) must necessarily be uncaused an without explanation.

As for the comparative simplicity of god and the universe, Occam’s Razor is a greatly overrated epistemic tool. If a scientist makes up a very simple theory which is untested or untestable, no one thinks that “it must be true because it’s simple”. (Actually, this issue is rather subtle, and I ought to write a post about this some time.)

Also, let me address the big bang. It doesn’t say the universe has existed for a finite amount of time. This is a very common misunderstanding of the big bang model. All the model really says is that ~13.7 billion years ago the universe was in a very hot dense state and that space has expanded since then. If you extrapolate a short way backwards in time from that hot dense state using the theory of General Relativity, you find that GR predicts a singularity, i.e. a “beginning of the universe”.
It is traditional to label this singularity t=0, and reference other events in the big bang from that point in time. This, however, is done merely for convenience. Actually, physicists are pretty certain that GR breaks down under the conditions which the big bang model postulates. So, that backwards extrapolation is actually not legitimate, and we should not attach any real significance to the point in time we labeled t=0.
To understand what happened in earlier on during the big bang (prior to the end of the Planck epoch), we need a theory of quantum gravity. And we don’t have one. So, maybe there was a beginning of time, or maybe things happened differently. We don’t know.

As for the Axiom of Choice, the meaning of “axiom” in logic and mathematics is subtly different from the word’s common usage. Here’s what Dictionary.com says:

axiom:
1. a self-evident truth that requires no proof.
2. a universally accepted principle or rule.
3. Logic, Mathematics. a proposition that is assumed without proof for the sake of studying the consequences that follow from it.

Mathematical axioms are not true or false (though they might be inconsistent with each other). When you use math to model the real world, what you do is make up some mathematical system (with any axioms you like) and then ask whether the world corresponds to the behavior of that abstract mathematical system. The axioms, and the abstract system arising from them are neither true nor false; the question is whether they successfully model the real world.

First off, let me just point out that current cosmology has a definite beginning to the universe (the Big Bang). I’m just saying.

But let us assume, for the sake of logical argument, that existence is infinite (which was the assumption att he time of Aquinas, actually). At that point, the universe is still contingent; i.e., it is possible that it could have never existed. With the universe being contingent by nature the fact that it exists, even in infinite form, means that there must be a necessary (’sufficient’) factor that caused its existence – as Aquinas said ‘and that we understand to be God’.

Adler phrased it like this;
1. The existence of an effect requiring the concurrent existence and action of an efficient cause implies the existence and action of that cause.
2. The cosmos as a whole exists.
3. The existence of the cosmos as a whole is radically contingent (meaning that it needs an efficient cause of its continuing existence to preserve it in being, and prevent it from being annihilated, or reduced to nothing).
4. If the cosmos needs an efficient cause of its continuing existence, then that cause must be a supernatural being, supernatural in its action, and one the existence of which is uncaused, in other words, the Supreme Being, or God.

Another linguistic explanation is put this way:
1. A contingent being exists (a contingent being is such that if it exists, it can not-exist)
2. This contingent being has a cause or explanation of its existence.
3. The cause or explanation of its existence is something other than the contingent being itself.
4. What causes or explains the existence of this contingent being must either be solely other contingent beings or include a non-contingent (necessary) being.
5. Contingent beings alone cannot cause or explain the existence of a contingent being.
6. Therefore, what causes or explains the existence of this contingent being must include a non-contingent (necessary) being.
7. Therefore, a necessary being (a being which, if it exists, cannot not exist) exists.

Swinburne’s argument by Induction on the Cosmological Argument is one that I, personally, don’t care for but that is quite popular with my peers. Let me quote a fairly good glodd of his argument;

“…if only scientific explanations are allowed, the universe would be a brute fact. If the universe is finite, the first moment would be a brute fact because no scientific causal account could be given for it. If the universe is infinite, each state would be a brute fact, for though each state would be explained by the causal conditions found in prior states plus the relevant physical laws, there is no reason why any particular state holds true rather than another, since the laws of physics are compatible with diverse states. That is, although the features F of the universe at time t are explained by F at time t1 plus the relevant physical laws L, and F at t1 is explained by F and L at t2, given an infinite regress there is no reason why F or L at tn might not have been different than they were. Since F and L at tn are brute facts, the same holds for any F explained by F and L at tn. Hence, regardless of whether the universe is infinite or finite, if only scientific evidence is allowed, the existence of the universe and its individual states is merely a brute fact, devoid of explanation.

The universe, however, is complex, whereas God is simple. But if something is to occur that is not explained, it is more likely that what occurs will be simple rather than complex. Hence, though the prior likelihood of neither God nor the universe is particularly high, the prior probability of a simple God exceeds that of a complex universe. Hence, if anything is to occur unexplained, it would be God, not the universe. On the other hand, it is reasonable to appeal to God as an explanation for the existence of a complex universe, since there are good reasons why God would make such a complex universe “as a theatre for finite agents to develop and make of it what they will””

Or, ‘taken at face value it is far more probable that God exists rather than the universe; since the universe obviously exists, it is highly probable that God exists as the cause of the universe’.

As for some of your other commentary – this proof was not calculated to describe the nature of the sufficient cause, just show the existence of one.

Also, your ’simplest’ version of the universe, an infinite existence, requires that the unprovable Axiom of Choice be true, meaning that a fair number of philosophers and mathematicians would disagree with you that this is the ’simplest’ version.

Secondly, your statement that the “infinite chain proposition means nothing within Logic” is rather bizarre; what Williams is trying to do is logically demonstrate that the infinite chain model (and, more generally, any model where all events are caused) is self-contradictory.

Lastly, if you think Williams’s argument against the infinite chain model (“…Outside of everything is nothing, and ‘from nothing, nothing comes’.”) is an example of clarity and rigor… well I just don’t know what to say to that.

The Causa Sui nature of the argument in full is very detailed; as I metioned earlier, I heartily recommend the shorter Summa. Of, simply peruse the bibliography from Williams’ page.

In short, the quantum infinite chain proposition means nothing within Logic.

Maybe I did misunderstand Williams; his argument is rather sloppy.
Can you produce a rigorous proof that the infinite chain model is self-contradictory? Do you have empirical evidence to show that the model makes predictions contrary to what we observe in the world?

You also seem to deeply misunderstand Williams’ statement. Williams does not refer to ‘the universe’, he refers to ‘all things’; this ,b/>included God. His argument is that the causal chain *includes* God as what St. Thomas called the “Prime Mover”. Since everything cannot be caused, that which is not caused is God. If you insist that there is no ‘uncaused thing’, this creates the paradox that there was no first effect (no Prime Mover) and, thus, there must be nothing *at all*.

So, again – the statement is logically valid. Causation exists, therefore there must exist an uncaused thing.