Gödel’s Ontological Argument [Draft] #

Overview #

Gödel’s proof is a formalization of Anselm’s idea (that God’s existence follows from the very concept of God), but cast in modal logic and second-order logic — with a heavy dose of mathematical elegance.

Instead of starting with “God exists,” Gödel starts with a very abstract idea:

Some properties are positive, and being God-like means having all positive properties.

Then the argument unfolds like this:

Key Ideas (in plain language) #

There are “positive properties.”
- Think of them as morally or metaphysically “good” — like omnipotence, benevolence, necessary existence.
- Gödel never defines what “positive” means — it’s left as an abstract primitive (like “axiomatically good”).
A God-like being has all positive properties.
- If a property is positive, then a God-like being has it.
- So: “God-like” = having every positive property.
Positive properties are necessarily positive.
- If something is positive, it’s not just incidentally so — it’s necessarily so, in all possible worlds.
Possibly, a God-like being exists.
- That is: in some possible world, there exists something that has all positive properties.
If it’s possible that a God-like being exists, then a God-like being exists necessarily.
- This follows from how he defines “God-like” and necessary existence as a positive property.
Therefore, a God-like being exists necessarily.
- The possibility of a God-like being’s existence forces it to exist in all worlds.

Gödel’s Argument (Step by Step with Symbols) #

Let’s walk through the core definitions and axioms, step by step:

Step 1: Define “Positive Property” #

Let P(ϕ) mean:

“ϕ is a positive property”

Gödel does not define what counts as “positive” — he just lays down axioms for how positive properties behave.

Axiom 1: A property is either positive or its negation is positive (but not both) #

P(ϕ) ⇔ ¬P(¬ϕ)

So if “being omniscient” is positive, then “not being omniscient” is not.

Axiom 2: Positive properties are necessarily positive #

P(ϕ) → □P(ϕ)

Positivity isn’t accidental — it’s modal, like a metaphysical truth.

Step 2: Define “God-like being” #

A being x is God-like, written G(x), if and only if:

For every property ϕ, if ϕ is positive, then x has it:
G(x) ⇔ ∀ϕ (P(ϕ) → ϕ(x))

So: God-like = has all positive properties.

Axiom 3: Being God-like is a positive property #

P(G)

That is, the property of being God-like is itself positive.

Step 3: Possibility of God #

From the above, Gödel proves:

◇∃x G(x) — it is possible that a God-like being exists.

(This is where the ontological argument really starts to take off.)

Step 4: Define “Necessary Existence” #

Let E(x) mean:

“x necessarily exists”

Formally:

E(x) ⇔ ∀ϕ (ϕ is an essence of x → □∃y ϕ(y))

We won’t get deep into essence, but roughly:

A property is an essence of x if:
- x has it, and
- it entails all other properties of x.

Gödel defines this carefully, and uses it to define necessary existence.

Axiom 4: Necessary existence is a positive property #

P(E)

This is the key: existence, when necessary, is a good thing.

Step 5: The explosion #

Given all the above, Gödel proves:

□∃x G(x) — A God-like being exists necessarily

Which, in modal logic, entails:

∃x G(x) — A God-like being exists

Why it seems so powerful #

It uses the machinery of modal logic, second-order quantification, and property theory.
Gödel’s proof is logically valid — formal and clean. It’s been computer-verified.
But it’s built on axioms about positivity and essence, which you may or may not accept.

Critiques and Controversy #

The logic is sound — but are the axioms justified?
Is “positive” a meaningful, objective notion?
Are we justified in assuming modal collapse doesn’t occur (i.e., all truths become necessary)?

There’s a famous result: Anderson and others showed that Gödel’s axioms lead to modal collapse — meaning:

If anything is true, then it’s necessarily true.

This undermines contingency — which many philosophers find unacceptable.

Wrap-up #

Gödel’s argument is a tour de force — an attempt to capture the idea of God as logically compelling under certain philosophical commitments.

It has no logical flaw — but it’s only as convincing as your acceptance of its axioms.