Can the existence of God be proven through possibility and necessity? Or should we be looking elsewhere?

Kurt Gödel (1906–1978) was a mathematician and philosopher best known for his incompleteness theorems, which reshaped modern mathematics. Less widely known is that he also developed a modal argument for the existence of God. Gödel never published this work himself; it remained in private notes that he shared with colleagues. One of them, the logician Dana Scott, later transcribed these notes, and the proof appeared posthumously.

Gödel’s proof relies on modal logic. Unlike classical logic — which evaluates whether statements are simply true or false — modal logic introduces operators for possibility (◇) and necessity (□). This framework allows philosophers to analyze different “possible worlds” and what could have been otherwise.

For example, in classical deductive logic we reason as follows:

All human beings are mortal. Socrates is a human being. Therefore, Socrates is mortal.

Modal logic, by contrast, lets us frame arguments in terms of possibility and necessity:

◇ “It is possible that unicorns exist.”

□ “It is necessary that 2 + 3 = 5.”

Gödel’s Argument

Following St. Anselm’s ontological proof for God’s existence, the argument faced substantial pushback. To situate Gödel’s approach, it is helpful first to consider Anselm’s version (which I discuss in my blog here). Among the most influential critics of Anselm was Immanuel Kant, who argued that existence itself is not a predicate. In other words, Anselm finds existence to be a quality that makes a thing greater. Kant, however, finds that existence itself is not a property that can be added onto something.

It was not until the 20th century that the discussion gained new traction, when Gödel reinterpreted the argument through the lens of modal logic. Gödel’s argument proceeds as follows:

Definition

Definition of God: God is defined as a being that possesses all positive properties.

Axioms

1. First Axiom: Either a property or its negation can be positive — not both.

• The first axiom means that if a property such as “omnipotence” or “moral goodness” is counted as positive, then its opposite (“not-omnipotence,” “moral badness”) cannot also be positive. Only one side of the pair can qualify as a genuine perfection. (If goodness is positive, then evil is not.)

2. Second Axiom: If a property is positive, then there is a possibility that it exists.

• The second axiom requires understanding Gödel’s intellectual background. He is drawing on the Scholastic lineage of René Descartes and Gottfried Wilhelm Leibniz, where a “property” is taken to mean a perfection. Within this framework, a positive property cannot be incoherent or impossible — otherwise it would not truly be perfect. If we were to list a so-called positive property that could never exist, we would in effect be declaring perfection itself to be impossible, which is self-contradictory. For example, imagine treating a “square-circle” as a positive property. A square-circle is logically impossible; no being could ever have this property. Calling it positive would entail that the term “positive” has no real meaning.

3. Third Axiom: Necessary existence is a positive property.

• The third axiom holds that necessary existence is itself a perfection. A being that exists necessarily — whose non-existence is impossible — is more perfect than a being whose existence is contingent, accidental, or could fail.

Theorems

1. If there is a possibility that God exists, then God must exist in a possible world.

• Per the first definition, God must possess all positive properties. And per the second axiom, there must be a possibility that positive properties exist in a possible world. So, at least in one world God exists.

2. If God exists in some possible world, then God exists in every possible world.

• Per the third axiom, one of God’s properties is necessary existence. This means that if God exists in even one possible world — and God is defined as the being that possesses all positive properties — then God must exist in every possible world, since necessary existence cannot be limited to just one world.

3. Therefore, God necessarily exists.

Criticisms of Gödel’s Ontological Argument

The most prominent criticism of Gödel’s ontological argument is that all contingent truths collapse into necessary truths. Jordan Howard Sobel’s work is central here: he shows that the axioms Gödel sets out require every truth to be necessary. If God is defined as the being with all positive properties and exists in all possible worlds, then there is no room for genuine contingency. For example, take the simple act of me drinking a matcha latte. Common sense says I could have chosen coffee instead — so this looks like a contingent truth. But under Gödel’s framework, if God necessarily exists in all worlds and grounds all properties, then the fact that I drank matcha must also be necessary. In other words, nothing could have been otherwise.

Though, it must be noted that modal collapse does not mean that Gödel’s argument is false. All it would mean is that necessitarianism is true (i.e., everything that exist must necessarily exist) with famous philosophers like Baruch Spinoza falling into this camp.

However, when it comes to Gödel’s argument, the most effective criticisms are those that strike at the foundational of the ontological argument. The proof depends on a set of definitions, axioms, and theorems that Gödel simply stipulates. If these assumptions are arbitrary, or if one rejects their plausibility, the entire structure collapses. In other words, the strength of the argument is only as strong as the acceptance of its starting points.

Ultimately, the question we should always ask ourselves is this: is this philosopher right? In Gödel’s case, what exactly does he get wrong? And if his framework fails, is there a better one available?