First order logic, set theory and God

[polymath257]

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Do we really have any reason to think that infinite phenomena have causes? All phenomena we have observed are finite, so that suggests that we only know about finite phenomena.

I'm not sure what you mean by "infinite phenomena." Are you thinking that a First Cause has to be infinite? 

I don't think we've gotten to that part of the argument yet.

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No, I'm suggesting that the axiom should state:
For all finite B, there is an A with A->B. It is possible that A=B.
All the effects we know about are finite. We have no basis for expacting that infinite systems have causes, Furthermore, for a first order system, you need axioms declaring how to form 'phenomena' like B.
In particular, the system V in your proof is likely to be infinite, so we need not expect it to have a cause. But even worst, in a first order set of axioms, it may not even be possible to talk about V.

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It might be an infinite regress. dron3 promises that Hatcher's argument avoids such a regress, but I haven't figured out how it does that yet, if it really does. 

Aquinas said he could not demonstrate that a per accidens series was not infinite in regress, but he did think he could show that a per se series must culminate in a first mover. So if you'd like an answer to your question you could work on those arguments. If it turns out that they're wrong, you'd at least be able to say why.

It does so by allowing the construction of the system V (a very large collection of events) in the proof. This is likely to be an infinite system, yet it has a cause.

In essence, it takes the whole of the infinite regress, if such exists, and declares that regress itself has a cause.

This is why P1 is suspect