The existence of logically undecidable statements leads us to a modern analogy of the Paradox of the Stone: Can God determine the validity of a logically undecidable statement? If ‘Yes’, then God is beyond logical reasoning. If ‘No’, then God is not all-knowing. Consequently, if God exists, then God is either beyond logical reasoning or God is not all-knowing. The latter is not compatible with Christianity, Islam, and other religions that account for well over half of the world’s population. In spite of the popularity of an all-knowing God, there is a contrary belief, finite theism, in which God has limited knowledge. We will stay with the over-whelming majority view and focus on an all-knowing God.

To avoid repeating ‘all-knowing God’, in the rest of this discussion ‘God’ will mean an all-knowing God, so that ‘belief’ will mean belief in an all-knowing God and ‘believers’ will mean believers in an all-knowing God. Hence, God is beyond logical reasoning for believers. It smooths over many problems for believers by making paradoxes (logically self-contradictory statements) inapplicable to God. However, it also prevents logical reasoning from being used to distinguish between competing concepts of God. This brings us back to God being primitive notion.

If God is beyond logical reasoning, then it is not possible to determine whether God exists using logical methods. This makes ‘God exists.’ a logically undecidable statement for believers. Some may find this to be heretical, but logical undecidability of ‘God exists.’ is actually essential for faith-based religions. If the existence of God could be proved or disproved, then the importance of faith would be greatly diminished, if not made irrelevant. Faith-based religions could then not exist in their present forms. Besides, the logical undecidability of ‘God exists.’ fits nicely in the mysterious, unknowable nature of God maintained by faith-based religions. 

What about non-believers? According to Bertrand Russell

“When you are studying any matter or considering any philosophy, ask yourself only: what are the facts, and what is the truth that the facts bear out. Never let yourself be diverted by what you wish to believe, but look only and surely at what are the facts.” 

By this standard (henceforth called Russell’s Criterion) the statement ‘God exists.’ should be considered logically undecidable, even by those who do not believe in God. To do otherwise would ignore centuries of trying to produce an acceptable proof of the existence or non-existence of God. Hence, ‘God exists.’ is logically undecidable for both believers and non-believers. 

Some may find the logical undecidability of ‘God exists.’ to be philosophically or theologically unacceptable. There are only three ways to avoid it: (1) reject consistent logical systems, (2) reject Russell’s Criterion, or (3) add an axiom concerning the existence of God. I will consider them in order. Rejecting consistent logical systems is too absurd to merit further comment. Much of philosophy tacitly relies on Russell’s Criterion, or something equivalent to it. There would be philosophical chaos if it is rejected. An axiom is a statement that is well-understood and non-paradoxical. This precludes undefined or ambiguous words, such as ‘God’, from being part of an axiom. In short, the logical undecidability of ‘God exists’ is unavoidable in any sensible setting. 

It needs to be stressed that the logical undecidability of ‘God exists.’ does not address whether God exists, only that the existence (or non-existence) of God cannot be established using logical arguments in a consistent logical system. Also, the logical undecidability of ‘God exists.’ does not imply agnosticism or atheism. People believe in the existence, or non-existence, of God for a variety of reasons that do not rely on logical reasoning. A discussion of these reasons is beyond the scope of this article. 

If faith is a firm belief in something for which there is insufficient evidence, then the logical undecidability of the existence of God makes belief, or disbelief, in the existence of God strictly a matter of faith. The existence of God being a matter of faith creates the paradox that God simultaneously exists (for some) and does not exist (for others), just as Schrödinger’s cat is simultaneously alive and dead. Detractors consider faith to be deluded (a term commonly used for something with which one disagrees) because it defies logic and reason. Of course, faith defies logic and reason. Otherwise, it would not be faith. But this does not mean that faith is necessarily deluded. Mahatma Gandhi was not deluded when he had faith that non-violent methods would lead to India’s independence. 

In summary:

1. If God exists, then God is either beyond logical reasoning or God is not all-knowing.
2. The statement ‘An all-knowing God exists.’ is logically undecidable.
3. Belief, or disbelief, in the existence of an all-knowing God is strictly a matter of faith.
   

My thoughts expressed above bear little resemblance to thoughts in the world around me. Most people conceive of God as being all-powerful and all-knowing, even those that do not believe in God. They could not care less about paradoxes. For them, God’s existence, or non-existence, is not a matter of faith. It is a matter of fact. The Bible or Qur’an is sufficient evidence of God’s existence for many believers. Genocides and natural disasters prove God does not exist for many non-believers. Each group considers it absurd not to believe as it does and summarily dismisses opposing beliefs as delusional. Collectively, believers and non-believers buttress my conviction that belief, or disbelief, in God is beyond logical reasoning and is strictly a matter of faith. “To one who has faith, no explanation is necessary. To one without faith, no explanation is possible.” (Thomas Aquinas)