Axiom or theorem?

Mathmatician Christian Calude has (quite cunnigly) asked "Is the existence of God an axiom or a theorem?"

It's the axiomatics that most seem to bother Richard Dawkins: "I do think that intelligent, sophisticated theologians are almost totally irrelvant to the phenomenon of religion in the world today...because they're outnumbered by vast hordes of religious idiots."

Yet does his belief that the non-existence of God is self-evident make him a bit of an axiomatic himself; an irreligious idiot?

Ruth Gledhill, the Times Religion Correspondent appears to have teased out the closet Pantheist in Professor Dawkins by chucking rather duplicitous vocab like numinous and transcendent at him.

"There is something our mind cannot grasp and whose beauty and sublimity reaches us only indirectly," he eventually volunteered. (Note though - Gledhill herself doesn't - that Dawkins does not mean by this something that one is obliged to worship.)

Ultimately the difference between Richard Dawkins's 'something' and the faith-heads' God is explanation. If Dawkins found God, he'd expect to understand Him, using step-by-step logic. (I'm being careful to capitalise here in case sic-os like Bill O'Reilly get on my case. See below.)

Trouble is, thanks to Gödel's incompleteness, Turing's incomputability and Heisenberg's uncertainty, we are already able to appreciate the likely role of the inexplicable in the fundamental fabric of our cosmic environment. God may not be compressible into His component axioms after all.

Anyway, when it comes to Science and Religion, it strikes me that there are essentially three positions that it is sensible to espouse (other than Gould's fence-sitting separate magesteria cop-out):

1) Current scientific theory and religion cannot be reconciled. Dawkins hints at this strongly but cannot prove it, and so ends up resorting to fundamentalist-type reasoning himself.

2) Current scientific theory can be reconciled with some sort of omnipotent, omniscient being, though perhaps not the the one described in the New and Old Testaments and the Koran.

3) Current scientific theory is wrong in as much as it disagrees with a literal interpretation of scripture.

A possible fourth position, that current scientific theory can be reconciled with pilars of stone, virgin births etc is not one that intelligent believers appear genuinely comfortable with, and as a result they will immediately drift into pantheistic sophisms the moment the subject comes up.