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Axioms and Definitions

As I am endeavoring to communicate in unequivocal terms in the work-in-progress Axioms (and as is already well known by mathematicians and logicians8), there is very, very little that can be considered to be a priori truth in the entire realm of human intellectual endeavor. Most of what was once considered a priori truth from the time of the early Greek mathematicians and geometers has slowly, painfully been revealed to be contingent truth, truth that can only be derived from unprovable axioms - assumptions that are necessary to the development of logical arguments. Modern mathematics is developed from sets of propositions that are no longer advanced as self-evident truth, but rather as just that - propositions. If you change the propositions you get a different, but quite possibly equally valid contingent theory.

For us to use reason in addressing questions about God, we therefore will require a common set of assumptions and a consistent set of definitions. Otherwise, with different assumptions or contradictory assumptions who could be surprised if one arrives at different conclusions (or any conclusion at all, given that one can prove anything from a contradiction)?

The theorem concerning God that this short work will state and prove therefore follows from a set of axioms and definitions, as any theorem must. This theorem concerns God and the Universe, so it seems suitable to start by precisely defining these two terms. Otherwise you could fail to understand the proof simply because the words used mean something different to you in your everyday discourse, perhaps because you speak a different language than English, perhaps because you simply never thought about their precise meaning.

The Universe: The Universe is defined to be everything that exists. This definition must be carefully qualified because we must not take the narrow viewpoint that everything that exists is the single space-time in which we appear to live. We must allow for the possibility of multiple space-time continua, or larger spaces in which our space-time is embedded. This is not to allege that we know whether or not such a larger space exists, only that when we use the term Universe we are referring to the union of all dimensions that have objective existence, known or unknown, simply connected, multiply connected, or disjoint, together with all of their contents. It is time-independent (obviously so, given that what we call ``time'' is a single dimension in our space-time continuum).

The point is that any assertion such as ``$A$ exists (or existed, or will exist) but is not a part of the Universe'' is a direct contradiction of the term Universe for any $A$. If a thing has objective being, it is in the set of all things that have objective being. Note well that this existential definition of the set itself completely avoids all difficulties with e.g. Russell paradoxes (for those that know what they are) as ``sets'' do not have objective being, they are merely figments of our imagination where the Universe is precisely the set of what is real whether or not we can correctly or precisely imagine it. Nor does this existential set have the problems associated with infinite Universal abstract sets for much the same reason - the Universe is precisely what it is, infinite or not, and our beliefs concerning it will not affect the reality in any way.

God: Our proof requires that we associate with the term God only four of the various qualities often associated with Deity. There may be other properties one wishes to assert, but in order to build a consistent rational theory of God, one must begin somewhere with a minimal set that everybody can agree on and then see if the others can be consistently asserted or asserted in a way that agrees with experience. The term ``God'' in this work refers to something that possesses that following qualities, that are common to nearly all monotheistic or monistic religious faiths

Unitary Being
God must have objective existence in order to be God, and not just ``be a myth'' or ``be an idea''. Also, there can be just one (or One, if you prefer) God at least with a capital G that has all of the properties we require.
God must possess a complete knowledge of the state of the Universe, right down to the last electron in the last space-time continuum, the last hidden dimension. For the moment we defer a discussion of just what ``knowledge'' is, but even without it we can agree that if God's knowledge of the Universe is incomplete - if there are things some otherwise very large and powerful being with objective existence doesn't know - then that being is not God, it is just a bigger and more powerful natural creature, existing within a partly unknown Universe, much like ourselves. We might well fear such a being. We might well obey it out of that fear. We would never worship it; indeed all free humans would hate it to the precise extent that it imposed its will by fiat upon us.
God must be ``everywhere''. In general this has been associated with omniscience - in order to know everything one must observe everything. In order to observe everything, one must be spatially contiguous in order to peform the observation. Of course any physicist knows that this is all very sloppy and incorrect - observation-derived knowledge is only meaningful in terms of entropy, which is missing information. We've already agreed that God has no missing information and is in a zero-entropy state relative to the Universe, which of course has consequences, one of them being the theorem we seek to state and prove. Nevertheless, omnipresence will be retained as a harmless and natural adjunct of omniscience and a reasonable property of God.
This is a tough one. It is usually interpreted as ``God can do anything God wants to'', but this is at once far too narrow and too broad (as was recognized even by the ``philosophers'' of the Church, e.g. Thomas Aquinas). God cannot make $\pi = 3$. God cannot possess contradictory properties or perform contradictory acts. God cannot make the impossible (defined as something that is logically contradictory) happen. However, this property is essentially irrelevant to my argument below - it is in fact a contingent corrolary of the only valid basis for information-theoretic omniscience and not really a separate property.

We will stop here. Most of the rest of the properties theistic religions ascribe to God - Creator of the Universe, omnibenificence, perfectly just, perfectly jealous, perfectly cruel, perfectly loving - are either (as we shall see) directly self-contradictory of the definitions and properties already listed above or fairly obvious anthropomorphic projections of human traits onto the Godhead (and are even more contradictory in combination with each other, as we contemplate an all-loving God who consigns self-aware creations to eternal fiery torment). Perhaps we will in the end have some room for more properties, perhaps not, but we must not introduce contradictions from the beginning.

Finally, we need a single assertion each from information theory and physics. From physics we need the idea that things that exist have properties. Properties we typically ascribe to things like quarks and electrons in physics (elementary particles out of which larger objects are built) include things like:

Things that exist, at least insofar as our direct experience goes, seem to have intrinsic properties, and in the case of so-called elementary particles, they are irreducible intrinsic properties.

The theorem below doesn't rely on any particular set of these properties, the independence or relationships between the properties, or whether or not we know what a complete set of the properties are, only that things that exist have them. It is valid whether or not we know what they are, it is valid however they might be ``causally'' entertwined and interrelated. It is valid whether or not they are classical or quantum properties or something else altogether.

I don't know what it would mean to say that a thing exists but has no properties at all - that it has no properties of its own and isn't made up of things that have properties of their own that it inherits in aggregate, that it cannot be correlated in any way with things that have properties (as that correlation would be a property). Such a thing would presumably ``exist'' at no place and at no time (in any generalized sense) and would in no way affect anything that did exist at some place and some time. I'd be pretty comfortable redefining the meaning of the word ``exist'' so that it excluded such a thing, if need be, but I don't think that it is.

We need to carefully bear in mind that coordinates we might use to symbolically represent those properties are not the properties themselves but rather our semantic or mathematical description of those properties. Given that the Universe is the set of all things that exist, and that things that exist have properties, and that properties can be represented in the abstract as coordinates, physics posits a coordinate description of the Universe that is the set of all of the coordinates corresponding to all of the properties of all of the things that actually exist.

This ``Universal set'' of coordinates is at its most abstract information, and we can use information theory to deduce certain very important conclusions concerning it and the Universe of actual things with actual properties that this coordinate description represents. We will need only one of these deductions below. We begin by mentally compressing the coordinate description - eliminating all redundant information. This leads us to an irreducible specification of the state of the Universe as a complete set of non-redundant coordinates (where a lot of the reduction will appear in the form of exact or approximate coordinate relationships, a.k.a. ``the laws of nature''). This information is self-encoded in the actual things that exist in the actual Universe.

We cannot represent this fully reduced coordinate description (which might well be infinite in size) within any system with less information content (where if infinite, we would require no less cardinality, since compressed it might well still be infinite in size). If we could, it wouldn't be fully reduced! Note that we are pretty safe in insisting on the existence of a maximum compression (or minimal representation of the information) because there is an information-theoretic strict lower bound of "no information" that is insufficient to reconstruct any non-empty coordinate description at all. There is no guarantee that our original, sufficient coordinatization of the state of the Universe is minimal, but somewhere between it and zero there must be one (possibly even more than one) that is minimal, a state of maximum compression.

That's all we really need below, I'm sorry I went so tediously about arriving at this simple statement but I wanted to make sure that it was clear.

next up previous contents
Next: The Pandeist Theorem Up: god_theorem Previous: A Purely Gratuitous Polemic   Contents
Robert G. Brown 2014-02-06