It is rare in the history of mankind that an actual theorem has been stated, let alone proven in a satisfactory way, concerning God. This lack of any empirically or rationally provable results has been one of the major stumbling blocks to rational religion, and of course that which is not, or cannot, be made rational remains irrational to the great dismay and mutual destruction of humanity.
This document proves, from simple premises and definitions consistent with the more or less standard model of God, an actual theorem concerning God and the Universe2 that might be of interest in the eternal conflict between those that allege that God ``created'' the Universe and those that allege that there is no God and that the Universe didn't require a creator. It does not resolve the question of whether or not there is a God, but it puts an end rather nicely to the question of whether or not any hypothesized God created a distinct and separate Universe. It is an ontological proof, but it avoids the fallacies associated with Anselm's Ontological Proof of God and doesn't require an axiom of infinity (although it can tolerate one) and thereby dodges some of the Gödelian Russell paradoxes of set theory as well.
It will become apparent as one studies and understands this theorem that it establishes strict constraints on theisms in order for them to be reasonable and hence possible. It does not address the deeper (and more difficult) issue as to whether or not any of theistic religion is plausible, as that requires further assumptions to be made and (as will be described in my book-under-construction Axioms) these assumptions invariably prove that God exists only by begging the question, by directly or indirectly including the conclusion in the premises. The axioms and definitions used below to prove this theorem, however, are ones that are frankly difficult to disagree with and still use English words in their commonly accepted (and nevertheless carefully defined) way, and do not beg the question addressed as the proof requires genuine synthesis from all of the axioms as well as input from various well-founded mathematical theories.
Let me be blunt: Most theistic religions of the world are not in accord with this theorem and hence are literally impossible. Because the theorem is based on reason (and its axioms are such that they can hardly be challenged and end up with a system of rational knowledge at all) this is simply too bad for those religions. They are incorrect. There is not the slightest window left open for them to be correct and have God possess certain key properties that those religions themselves ascribe to God. Augustine, Aquinas or Anselm, ardent Christian theists that they were, would have to accept this conclusion even as all of their arguments for a Christian theistic ``Creator'' deity (such as asserting that a God that is real is greater than any imaginary God so that somehow my imagination of God becomes a proof of God) have long since been shown to be fallacious and even somewhat silly as e.g. General Semantics has carefully drawn lines between ``things in our imagination'' (the map) and ``things that have objective existential being'' (the territory) where the former is most certainly not identical to or in any sort of logically necessary one to one correspondance with the latter.
As to why the theorem proven below has been missed by all of the world's philosopher, the answer is simple. First of all, it really hasn't; the assertion itself is the basis for a whole family of the world's religions3 and a common heresy of many of the newer ones. The assertion played an important role in the Enlightenment of Europe and in the establishment of the United States. It was the directly stated belief of Franklin, Jefferson, Thomas Paine, George Washington, and many other of America's founding fathers and important philosophers around the globe.
The assertion has thus been around for a very long time - only the proof has been lacking. The proof given below is (as far as I know) unique and has only been possible for the last few decades, and then only accessible to someone with an interest in and knowledge of philosophy, religion, physics, mathematics, and computer science as elements are drawn from all of these disciplines to make the argument.
Second, most philosophical reasoning about God has focused on ``proving'' or ``disproving'' God's existence using reason, a process that has continued long after Hume's correct and unassailable observation that such a proof is impossible - God cannot be proven by (reasonable) inference as finite observations can never suffice to prove the infinite, and God cannot be deduced without begging the question in the axioms used in the proof. The inexorable progress of science in closing the gaps in human knowledge has systematically reduced the empirical plausibility of any model of a God that intervenes in the Universe in detectable ways - there is simply no reliable, systematic body of evidence that suggests that this occurs.
This theorem treats the question of God's existence as being fundamentally unprovable, empirically unsupported, but possibly true anyway. This possibility is allowed by and logically consistent with Gödel's theorems, wherein there exist true - or false - but unprovable theorems in any sufficiently complex system of formal reason, and of course empirically absence of evidence is not necessary or sufficient evidence of absence, it merely reduces the plausibility of the assertion given the evidence in a purely Bayesian sense, especially given competing explanations that are well supported and leave little room.
To deliberately avoid being bogged down in the mire, then, the assertion proven below is a simple conditional statement - if God exists, then certain conclusions can be drawn concerning Its nature. No attempt is made to use these conclusions in reductio ad absurdem to prove or disprove the condition because post-Hume we know a priori that this is impossible. The conditional theorem thus proven leaves the possibility of rational belief (in God) in perfect balance with rational disbelief as being one of the questions that cannot be answered with certainty by any finite being, while strongly constraining the set of admissible consistent theistic systems contingent upon axiomatically asserted belief.
Why should we bother to state and prove such an obviously ``dangerous'' theorem in a world already pulled apart by religious conflict? On a daily basis individuals commit murder and acts of war in the name of their favorite brand of supposedly divinely inspired perfectly true theistic scripture. This theorem proves that the detailed descriptions of the properties assigned to God in these scriptures to be not merely unlikely, not merely unsupported by positive evidence and contradicted by much negative evidence, they are a priori impossible. They assert one property in particular - the assertion of an essential dualism between a ``standard model'' God and Everything Else - that is essentially self-contradictory.
One can prove anything from inconsistent axioms, from a contradiction, and so it has proven throughout human history where every possible variant of human faith has spoken of God's love and in the same breath asserted God's wrath, has spoken of God's perfect justice and followed it up with a description of God's supposed actions that any child could see are perfectly unjust, has spoken of God's desire for peace and goodwill and followed it with war, persecution, torture, murder, slavery, greed all in the name of that God. While religion can be a source of good, it is all too often a source of genuine evil in the world as people fight over a vision of deity that is now demonstrably false.
It is a fundamental and empirically supported belief of the author of this work that the world is ill-served by unreason, by deliberate self-delusion on the part of each and every human alive. Humans cannot, through the empirical process or for that matter through pure logic, arrive at perfect, unconditional knowledge of any truth - we simply build the best, most consistent web of provisional knowledge that the evidence of our senses and common sense can manage. We quite literally should believe that which it is best to self-consistently believe, given the unbiased evidence and the rest of the interlocking network of beliefs. This is precisely the basis of the scientific worldview, which is infinitely open-minded about non-contradictory assertions, but which requires sound reasons and consistency with other soundly held beliefs in order to assign any signficant probability or plausibility to an assertion.
It is not the basis of scriptural theistic religions, which positively thrive on anecdotally reported implausible assertion after assertion in their myths, legends, creeds and tenets. Yet because these theisms admit pure magic as a causal agency in the Universe - basically anything at all can happen merely because God wills it so - no possible contradiction between their myths, however absurd, and science will suffice to convince them that their core mythology and derived creed is false! For that reason a proof that demonstrates a logical inconsistency is infinitely precious.
The world is just completing the first of the revolutions begun with the Enlightenment. The Age of Kings is over, and freedom from individual or hegemonic rule to a greater or lesser extent is now nearly universal around the world. Never before have the people of the world been more free to shape the mutual destiny of humanity in constructive ways, guided by human reason. The second revolution is well underway - the liberation of the human species from literal belief, enforced by threats and extortion, in complex Bronze Age mythologies that distort the worldview and judgement of all their adherents and that today are very nearly the only significant cause of large scale human misery outside of impersonal natural disasters and the equally natural accidents of life. Given all of the evidence of falsehoods in their theistic scriptures plus the proof that their core belief system isn't just contradicted by evidence, it is logically impossible, perhaps the world can at last carry this revolution through to a bloodless conclusion, by simply convincing a single generation of to reject the inconsistent and demonstrably false scriptural theisms of their parents and/or society in favor of any of the beliefs systems that are consistent with the theorem below: atheism, pandeism, a modern secular Buddhism, scientific rationalism.
None of these worldviews have a salvation meme or a damnation meme. They leave humanity free to decide its own destiny without any threat but ourselves and nature. If anything, they obligate good and socially constructive behavior more strongly than any of the theisms, because The Good is no longer defined to be ``believing what this scripture tells you to believe because of its great authority'', it is ``believing what makes the most sense''. It is pure common sense for all enlightened secular humans to seek a peaceful and safe world, to engineer a world that maximizes fairness and personal human freedom, that provides challenges for those endowed with greater abilities and protections for those unfortunate enough to have lesser ones.
To make this point inescapably clear, the next section is a strong argument - a veritable polemic - against authority-based scriptural theisms in general, those of the Abrahamic faiths (which are largely responsible for the remaining upheaval and war in the world) in particular.