A Theodicy of Incompleteness
[The following is an excerpt from an article originally printed in Hope's Reason: A Journal of Apologetics and reprinted as the first chapter of my book Transcending Proof.]
Like most of the great mathematical discoveries by the great mathematicians, the famous incompleteness theorems published by Kurt Gödel in 1931 almost completely escape the comprehension of the average man on the street. Nonetheless, scholars familiar with the work of Gödel and his theorems have gone to the trouble of translating his texts – not only from the original German, but from the abstract language of logic and high-level arithmetic. What they describe is a powerful insight with profound limiting implications for otherwise seemingly unbounded areas of research, such as artificial intelligence and theoretical cosmology. I suspect they also have implications for theodicy.
Using sophisticated mathematical and logical machinery, Gödel managed to prove with the incompleteness theorems that in most any formal and consistent axiomatic system, there will be a true statement derivable from the system which nonetheless cannot be proven within the system. The statement in question can be proven in principle (as it is true), through the addition of more axioms, but this expansion results in a larger system in which the principle of incompleteness again holds: New statements will be derivable from the new system, which cannot be proven within the new system.
To illustrate the theorem I will take the liberty to borrow an analogy from Rudy Rucker, that of a truth machine which houses all known truth and can answer all questions asked of it with only true statements. A truth machine operator approaches the machine and types in the following sentence:
"The truth machine will never say that this sentence is true."
Then the operator asks the machine if the above sentence, as stated, is true or false. If the truth machine decides the sentence is true, it cannot say so (because the sentence states that the truth machine will not say it is true). If the truth machine decides the sentence is false, then again it cannot say so (because it only answers with true statements) – yet its failure to say so is precisely what the sentence says of the truth machine. It is true, then, that the truth machine will never say that the sentence is true. Though true in itself, the undecidability of the sentence for the truth machine means that its truth cannot be recognized by that same machine.
All this implies that as outside observers, we can somehow ascertain a truth that even a perfectly programmed truth machine cannot. This implies in turn that we, along with this special insight that only we can see, in some sense transcend any programmed system – even a system that houses all known truth. How can this be? Well, for one thing we have not been programmed. Human beings are evidently not reducible to machines, any more than our thoughts are reducible to abstract statements derived from formal systems of logic or mathematics. Often the undecidable statement in a proof of Gödel's theorem is termed self-referential, and this is telling; for what a machine lacks by its classical definition is self-awareness. Penrose argues that with this ability to reflect human beings alone can see both sides of a paradox, whereas a machine can only process inputs given it from outside itself. In a brilliant stroke of genius eminently logical and equally paradoxical, Gödel managed to establish the critical distinction between God-given reason and mechanical computation.
Technically Gödel's theorems only hold in the context of consistent systems featuring formal language, system-specific axioms, and rules of inference. Peano Arithmetic is thought to be the ideal such system. Euclidean geometry is also said to suffice. But the principle appears to apply more generally. For example, Stephen Hawking has argued that the eclipse of classical Newtonian physics by the mutually incompatible theories of quantum mechanics and general relativity suggests incompleteness of the physical universe. Though mathematical models can be created which approximate the fundamental structure of the universe, they cannot be proven in principle because human observers are entities within the very system under observation:
But we are not angels, who view the universe from the outside. Instead, we and our models are both part of the universe we are describing. Thus a physical theory is self-referencing, like in Gödel's theorem. One might therefore expect it to be either inconsistent or incomplete. The theories we have so far are both inconsistent and incomplete.
Even more so, theological explanations for evil in a physical universe whose theories are inconsistent or incomplete should be expected to appear similarly inconsistent or incomplete. Pressing the idea yet further, Thomas Nagel maintains that in light of the unavoidable subjectivity of human perceptions, "any objective conception of reality must acknowledge its own incompleteness."
A less formal but no less baffling undecidable statement facing any theodicy project might go something like this: "God's act of creating humans free to choose between good and evil is morally justifiable." If we say that the sentence is true, we imply that the freedom to choose evil is morally justifiable (though evil by definition is not morally justifiable). If we say that the sentence is false, we imply that the freedom to choose good is not morally justifiable (though good by definition is morally justifiable). The former means leaving the floodgates open to various forms of evil and its painful consequences. The latter means closing the door to love, friendship, adventure, growth, discovery, and personal accomplishments – in short, an absence of any meaningful experience of good. Either situation could rightly be described as evil. Theodicy in one sense remains woefully incomplete.
In another sense, however, the Scriptures supply a complete and coherent solution to the problem of evil. As Eleonore Stump suggests, certain Christian beliefs speak uniquely to the problem of evil: The fall of Adam (and by extension all of humanity); the onset of natural evil ("a curse upon the earth") through Adam's fall; and the eternal destination of either heaven or hell awaiting all people, depending on the state of their relationship to God, principally through faith in Christ (or willful lack thereof). Indeed, a thoroughly biblical Christian response to evil alone seems capable of answering the questions still confronting us:
1. How can God create an eternal paradise, given the priority he places on moral free will?
2. Why has God not already created an eternal paradise complete with morally free beings, given that he has the ability to do so? (Or, why is this-worldly existence even necessary?)
These questions really turn on one another. God can create an eternal paradise featuring sheer moral goodness only if its inhabitants are free to choose the good. But such a paradise requires that its inhabitants never choose evil, which implies a restriction on freedom. Just what is it, then, that makes it possible to retain human volition and at the same time ensure uncorrupted goodness? Jesus preached the answer consistently: the coming of the kingdom of God. The theology of the kingdom, especially its eschatological and eternal aspects, depicts a gradual but final and irreversible, i.e., complete, triumph of good over evil. As Jesus preached it and as most New Testament scholars acknowledge, the kingdom can be best viewed as having already arrived in one sense and yet awaiting its complete fulfillment in another. This is the "Already-Not Yet" paradigm, which suggests incompleteness in theology.
From this perspective, the creation of the world as described in Genesis was not the end of God's work of creation, but only the start of a much more expansive creative-redemptive program with ultimate, everlasting joy in view. This creative-redemptive program, as I have called it, consists of three distinct phases. During the first phase in the paradise of Eden, human free will was unrestricted with respect to choosing among certain moral and relational options. Among the numerous fruit-bearing trees in the garden were both the tree of life and the tree of the knowledge of good and evil. Despite God's warning that death would result, Adam (following Eve) ate of the fruit of the knowledge of good and evil; and of course to know good and evil is to know evil, and to know evil is to experience it. Given the basic truth of the doctrine of original sin or universal depravity, that all men have shared significantly in the transgression of Adam, all humans have experienced evil directly both as perpetrators and victims.
Inhabitants of a world fallen and cursed by sin, we are now in the second phase of God’s creative program. Having partaken of the knowledge of good and evil, we still operate with free will but with the added "advantage," so to speak, of being better (but still not completely) informed. Experience has taught us, i.e., Christian believers, that sin breeds more pain than pleasure in this life, and death at the end of it. Equally we have tasted of the forgiveness of sins, the liberating life of God in Christ, and the comforting ministry of the Holy Spirit. For believers, then, the innate human appetite for evil has been weakened and becomes ever weaker with our growth in the faith. Replacing that old craving for transitory pleasure is a desire for the eternal knowledge of God himself, the very source of all good things. On such a view, this-worldly existence is necessary as the arena in which eternally binding choices are made, and where evil – especially the irrational, excruciating sort we prefer to call pointless and gratuitous – serves as a powerful inducement to seek God rather than sin. "So things that contribute," says Stump, "to a person's humbling, to his awareness of his own evil, and to his unhappiness with his present state contribute to his willing God's help." She then concludes that "moral and natural evil make such a contribution." Jesus said simply, "Blessed are the poor in spirit, for theirs is the kingdom of heaven." In a fallen world, if no other, we are able to hear, freely and clearly, the divine call to repentance from sin and ongoing faith in Christ. Evil in that case might not be a senseless aberration from God's creative-redemptive plan, but an essential part of it.
Nonetheless, the third phase of God's creative-redemptive design alone will bring about the completeness we seek. Only in the future, final consummation of God’s plan will we realize how one can remain ever free to love God and others but never free to become evil. Although the logical compatibility of evil and divine benevolence, of free will and eternal blessedness, cannot be strictly proven within the system of this world, Scripture posits its provability in the larger transcendent system of the kingdom. In the eternal kingdom of heaven theodicy will be completed. But of course no theodicy will be necessary. God will wipe every tear from our eyes and every trace of evil will have vanished away forever, not in violation of our free will, but in the divine response to it. This may explain why there is no tree of knowledge of good and evil in the heavenly paradise of Revelation 22 – only a tree of life. Having already tasted the bitter fruit of the knowledge of good and evil, and as a result having freely renounced sin and embraced eternal life in Christ by faith, we will enter the New Jerusalem prepared to joyfully partake of the tree of life forever. Only then and there, in the eternal kingdom of heaven, will we experience the culmination of both genuine freedom and everlasting joy.
 The idea goes something like this: For any system based on formal language L, there will be a self-referentially true statement G coded in L such that neither G nor not-G is provable in the system. G, then, is true but formally undecidable. Either the system is incomplete with respect to the truth of G, or the system is inconsistent (consistency here means that in principle no statement can be derived that is both proven and disproven via the axioms of the system). But since G is true (as can be proven in principle by expanding upon the system to include true axioms bearing on the truth of G), the system must be incomplete with respect to the truth of G.
 The illustration as I describe it is a condensed and modified version of Rucker's step-by-step explanation depicting a "Universal Truth Machine" and Gödel himself as its operator in Rudy Rucker, Infinity and the Mind (New York: Bantam Books, 1982), p. 174.
 "Reflection principles provide the very antithesis of formalist reasoning. If one is careful, they enable one to leap outside the rigid confinements of any formal system to obtain new mathematical insights that did not seem to be available before." – Roger Penrose, The Emperor's New Mind (New York: Oxford, 1989), p. 144.
 Stephen Hawking, "Gödel and the End of Physics," lecture given at the Dirac Centennial Celebration, Cambridge, UK, July 2002, http://www.hawking.org.uk/godel-and-the-end-of-physics.
 Thomas Nagel, The View from Nowhere (New York: Oxford University Press, 1986), p. 26.
 Eleonore Stump, "The Problem of Evil," Faith & Philosophy, Vol. 2, No. 4 (Oct. 1985), p. 398.
 For a comprehensive survey of historical and contemporary theology of the kingdom of God, see Mark Saucy, The Kingdom of God in the Teaching of Jesus (Dallas: Word, 1997).
 Stump, "The Problem of Evil," p. 409.
 Matthew 5:3, New King James Version. The New Century Version describes these poor as "they…who recognize their spiritual poverty." Presumably they belong in the same spiritual category with those who mourn, the meek, those who hunger and thirst for righteousness, the merciful, the pure in heart, the peacemakers, and those who are persecuted for righteousness' sake, Matt. 5:4-10.