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Towards a Logical Understanding of Paradox: Gödel’s Theorem and God’s Truth

By Noel Weichbrodt

Jesus said, “Blessed are the poor in spirit, for theirs is the kingdom of heaven” and “Blessed are the meek, for they will inherit the earth.”¹ Lao-Zi said, “Be twisted and one shall be whole. Be crooked and one shall be straight. Be hollow and one shall be filled…Have little and one shall obtain. But have much and one shall be perplexed.”² Mumon said

The student Doko came to a Zen master, and said: “I am seeking the truth. In what state of mind should I train myself, so as to find it?”
Said the master, “There is no mind, so you cannot put it in any state. There is no truth, so you cannot train yourself for it.”
“If there is no mind to train, and no truth to find, why do you have these monks gather before you every day to study Zen and train themselves for this study?”
“But I haven’t an inch of room here,” said the master, “so how could the monks gather? I have no tongue, so how could I call them together or teach them?”

“Oh, how can you lie like this?” asked Doko.

“But if I have no tongue to talk to others, how can I lie to you?” asked the master.

Then Doko said sadly, “I cannot follow you. I cannot understand you.”

“I cannot understand myself,” said the master.³

Perhaps I will have convinced you by the end of this paper that we may add to the above list “Gödel said, ‘Any sufficiently complex formal system is either incomplete or inconsistent.’” All four quotes posit paradoxes as the way to dialectically approach truth. All four work from their respective disciplines and speak to their respective disciples.

Mumon was a Zen Buddhist teacher who collected sayings like this one, called koans, that attempt to break the mind of logic so it may become Buddha. Lao-Zi was a great Daoist teacher, in whose beliefs we can see the “grappling with the intuition of a transcendent origin which they recognize to be beyond all human defining, description, and grasping…language falls silent before it.”⁴ And Jesus was God’s Son, who came from heaven to earth to give instruction to his followers and to die for the sins of each person, eternally.

These teacher’s teaching are not rationalist teachings, for they appeal to a sort of intuition and instinct in humans to try to wrestle with incompatible truths in their mind and thereby find themselves no resolving the truths, but living the life the truths call them to live. Truth is not self-defined, but is contained in people. In the Daoist case, truth is found in the Dao, a negation of anything that is approached by the person emptying themself. In Zen Buddhism, it is found in the Buddha, which is the transcendent being that we are all finding. In Christianity, truth is found in the person of Jesus.⁵ It is the Christian view of paradox that I will be interacting with, but understanding the other two views above helps to gain an appreciation for the wide appeal to paradox in human thought.

Paradox runs through the Christian faith. Appendix I quotes a few passages from Scripture that present paradoxes as truth. Kierkegaard, Barth, and other neo-orthodox theologians and believers “believed theological assertions of the faith to be paradoxical,” requiring believers “to hold opposite ‘truths’ in tension.”⁶ Calvin and other Reformers said that there are secret decrees⁷ of God that we cannot know. “It is foolish and rash to inquire into the unknown more deeply than God allows us to know.”⁸

Jesus himself presents us with a paradox: infinite God becomes finite man. One man died for the sins of the whole world, past and future. And what Jesus preached in the span between those two paradoxes is itself a paradox. The last will be first, he said. The humble will be exalted. The wise will be shown foolish. Lose your life so you may have life. Believe what you cannot see.

Teachings such as these point to an incompleteness in human understanding. This incompleteness is what allows such paradoxes to be called “true”, without being able to explain how or why the paradoxes are true. Intuitively, we know we must affirm the paradox, but we cannot resolve the paradox by ourselves. That the paradoxes of Christianity are divine in origin gives us further reason to believe that they must be affirmed as true, and explored as truth should be explored, but are irreconcilable to us.

In the same way, logic is incomplete and must affirm truths that are either contradictory or cannot be proven. Kurt Gödel proved this in his 1931 paper “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”.⁹ What Gödel did was make elementary mathematical number theory say “G is not a theorem of this system”, showing that there are true statements in number theory, and by extension, any high-level formal system (e.g. logic), that cannot be known within the system.¹⁰

There are two aspects of Gödel’s Theorem that are important to our discussion. The first, already discussed, is that any formal system, and by extension most everything in the world concerned with finding truth, cannot find all truths. All of our ways of finding truth depend on rationally searching out and finding truth. The ultimate expression of this rationality is found in logic. Logic systematizes human reason into something that can be manipulated symoblically to present to our reason different truths that logic can claim it has found to be true. And our reason, since it has decided that logic is reason’s fullest expression, assents to logic’s truth-claims. But these truth-claims of logic are not complete, and it follows that reason is in some important way incomplete as well, unable to capture all truths and put them into the human mind.

The second aspect of Gödel’s Theorem is that there does exist a complete understanding of truth—somewhere beyond our reach. In Gödel’s paper, we know that the statement G is true because it is true in elementary number theory. But once the elementary number theory starts talking about itself, G no longer becomes provable in the same elementary number theory. But we know G is true! We have a higher-level understanding that enables us to look back into number theory and say that G is true, even though the system cannot say G is true. Hofstadter notes “Though none of us will ever be able step back far enough to see the ‘big picture’, we shouldn’t forget that it exists.”¹¹ Similarly I suggest, God holds all truth in this divine, higher-level hands.

This is why I am comfortable, as a rational being, with the paradoxes of Jesus. I do not need to appeal to a non-rational justification for my faith, because reason itself dictates that it has limits on what it can know, and parts of my faith lie outside of those limits. Hofstadter notices this when he says “…all results essentially dependent on the fusion of subject and object [e.g. logic, math, physics] have been limitative results.”¹² Historically, the rationalist challenge to the Christian faith has been met—not by a reordering of the Christian faith, but by a reordering of reason.

That logic is incomplete suggests that there is a higher-level understanding of logic that we are incapable of accessing, a higher-level that would loop back to the level of our logic and explain to us why our logic is incomplete. This higher-level understanding is God’s understanding. To God, the paradoxes of Jesus are not paradoxes—they are resolved into a whole, complete, consistent truth.

Bibliography

Holy Bible, Revised Standard Version.

The Spirit of Chinese Philosophy: An Introduction, an unpublished paper by Dr. Ling-Mei Petcher (neé Lim) presented to the Covenant College Philosophy Club 23 April 2002.

Gödel, Escher, Bach: an Eternal Golden Braid by Douglas Hofstadter, Twentieth-anniversary Edition. Published by Basic Books in 1999.

Evangelical Dictionary of Theology Second Edition, edited by Walter A. Elwell. Published by Baker Academic in 2001.

Psychopannychia by John Calvin.

“On Formally Undecidable Propositions of Principia Mathematica and Related Systems I” by Kurt Gödel.

Appendix 1: Paradoxes in the Bible

Jesus

John 1:1-14

   In the beginning was the Word, and the Word was with God, and the Word was God. He was in the beginning with God; all things were made through him, and without him was not anything made that was made. In him was life, and the life was the light of men. The light shines in the darkness, and the darkness has not overcome it. There was a man sent from God, whose name was John. He came for testimony, to bear witness to the light, that all might believe through him. He was not the light, but came to bear witness to the light. The true light that enlightens every man was coming into the world. He was in the world, and the world was made through him, yet the world knew him not. He came to his own home, and his own people received him not. But to all who received him, who believed in his name, he gave power to become children of God; who were born, not of blood nor of the will of the flesh nor of the will of man, but of God. And the Word became flesh and dwelt among us, full of grace and truth; we have beheld his glory, glory as of the only Son from the Father.

Matthew 5:1-11

   Seeing the crowds, he went up on the mountain, and when he sat down his disciples came to him. And he opened his mouth and taught them, saying:

   "Blessed are the poor in spirit, for theirs is the kingdom of heaven.

   "Blessed are those who mourn, for they shall be comforted.

   "Blessed are the meek, for they shall inherit the earth.

   "Blessed are those who hunger and thirst for righteousness, for they shall be satisfied.

   "Blessed are the merciful, for they shall obtain mercy.

   "Blessed are the pure in heart, for they shall see God.

   "Blessed are the peacemakers, for they shall be called sons of God.

   "Blessed are those who are persecuted for righteousness' sake, for theirs is the kingdom of heaven.

   "Blessed are you when men revile you and persecute you and utter all kinds of evil against you falsely on my account.

Matthew 10:38-39

   and he who does not take his cross and follow me is not worthy of me. He who finds his life will lose it, and he who loses his life for my sake will find it.

Matthew 16:24-26

   Then Jesus told his disciples, "If any man would come after me, let him deny himself and take up his cross and follow me. For whoever would save his life will lose it, and whoever loses his life for my sake will find it. For what will it profit a man, if he gains the whole world and forfeits his life? Or what shall a man give in return for his life?

Matthew 19:29-30

   And every one who has left houses or brothers or sisters or father or mother or children or lands, for my name's sake, will receive a hundredfold, and inherit eternal life. But many that are first will be last, and the last first.

Matthew 20:1-10

   "For the kingdom of heaven is like a householder who went out early in the morning to hire laborers for his vineyard. After agreeing with the laborers for a denarius a day, he sent them into his vineyard. And going out about the third hour he saw others standing idle in the market place; and to them he said, 'You go into the vineyard too, and whatever is right I will give you.' So they went. Going out again about the sixth hour and the ninth hour, he did the same. And about the eleventh hour he went out and found others standing; and he said to them, 'Why do you stand here idle all day?' They said to him, 'Because no one has hired us.' He said to them, 'You go into the vineyard too.' And when evening came, the owner of the vineyard said to his steward, 'Call the laborers and pay them their wages, beginning with the last, up to the first.' And when those hired about the eleventh hour came, each of them received a denarius. Now when the first came, they thought they would receive more; but each of them also received a denarius.

Matthew 20:15-16

   ”Am I not allowed to do what I choose with what belongs to me? Or do you begrudge my generosity?' So the last will be first, and the first last."

Mark 9:34-35

   But they were silent; for on the way they had discussed with one another who was the greatest. And he sat down and called the twelve; and he said to them, "If any one would be first, he must be last of all and servant of all."

Paul

1 Corinthians 1:19-21

   For it is written, "I will destroy the wisdom of the wise, and the cleverness of the clever I will thwart." Where is the wise man? Where is the scribe? Where is the debater of this age? Has not God made foolish the wisdom of the world? For since, in the wisdom of God, the world did not know God through wisdom, it pleased God through the folly of what we preach to save those who believe.

1 Corinthians 2:1-7

   When I came to you, brethren, I did not come proclaiming to you the testimony of God in lofty words or wisdom. For I decided to know nothing among you except Jesus Christ and him crucified. And I was with you in weakness and in much fear and trembling; and my speech and my message were not in plausible words of wisdom, but in demonstration of the Spirit and of power, that your faith might not rest in the wisdom of men but in the power of God. Yet among the mature we do impart wisdom, although it is not a wisdom of this age or of the rulers of this age, who are doomed to pass away. But we impart a secret and hidden wisdom of God, which God decreed before the ages for our glorification.

1 Corinthians 13:1-2

   If I speak in the tongues of men and of angels, but have not love, I am a noisy gong or a clanging cymbal. And if I have prophetic powers, and understand all mysteries and all knowledge, and if I have all faith, so as to remove mountains, but have not love, I am nothing.

1 Corinthians 15:49-53

   Just as we have borne the image of the man of dust, we shall also bear the image of the man of heaven. I tell you this, brethren: flesh and blood cannot inherit the kingdom of God, nor does the perishable inherit the imperishable. Lo! I tell you a mystery. We shall not all sleep, but we shall all be changed, in a moment, in the twinkling of an eye, at the last trumpet. For the trumpet will sound, and the dead will be raised imperishable, and we shall be changed. For this perishable nature must put on the imperishable, and this mortal nature must put on immortality.

Romans 8:29-30

   For those whom he foreknew he also predestined to be conformed to the image of his Son, in order that he might be the first-born among many brethren. And those whom he predestined he also called; and those whom he called he also justified; and those whom he justified he also glorified.

Romans 11:32-35

   For God has consigned all men to disobedience, that he may have mercy upon all. O the depth of the riches and wisdom and knowledge of God! How unsearchable are his judgments and how inscrutable his ways! "For who has known the mind of the Lord, or who has been his counselor?" "Or who has given a gift to him that he might be repaid?"

Moses

Deuteronomy 29:29

The secret things belong to the LORD our God; but the things that are revealed belong to us and to our children for ever, that we may do all the words of this law.

¹ Matthew 5:3,5 RSV

² From the Dao De Jung by Lao-Zi. Taken from The Spirit of Chinese Philosophy: An Introduction, an unpublished paper by Dr. Ling-Mei Petcher (nee Lim) presented to the Covenant College Philosophy Club 23 April 2002. p.8.

³ Gödel, Escher, Bach: an Eternal Golden Braid by Douglas Hofstadter, Twentieth-anniversary Edition. Basic Books, 1999. p.250-251

⁴ The Spirit of Chinese Philosophy, p. 8

⁵ John 1. Recall the original Greek word for “Word”, “Logos”.

⁶ “Neo-orthodoxy” by R. V. Schnucker in the Evangelical Dictionary of Theology Second Edition, edited by Walter A. Elwell. Published by Baker Academic, 2001.

⁷ See Deut. 29:29 in Appendix I

⁸ Psychopannychia by John Calvin.

⁹ “On Formally Undecidable Propositions of Principia Mathematica and Related Systems I” by Kurt Gödel, published in 1931. Available on the web at http://www.ddc.net/ygg/etext/godel/godel3.htm

¹⁰ For background on the development of Gödel’s Incompleteness Theorum, I am biased to my paper, “Gödel and the Death of Modernist Metamathematics”, available on the web at http://www.weichbrodt.org/text/godel.html

¹¹ Gödel, Escher, Bach. p. 710

¹² Ibid. p. 699.