God exists!

Gödel’s ontological proof is a formalization of the ontological argument for the existence of God, originally developed by Anselm of Canterbury, using modal logic. Kurt Gödel, a renowned logician, constructed a rigorous proof leveraging mathematical logic to express the argument. The proof uses modal logic with axioms and definitions to argue for a necessary being with “god-like” properties.

Gödel’s Ontological Proof

Gödel’s proof is based on modal logic, where \(\Box\) denotes necessity (“true in all possible worlds”) and \(\Diamond\) denotes possibility (“true in some possible world”). The proof defines a “god-like” being as one possessing all positive properties necessarily and shows that such a being must exist in all possible worlds.

Definitions and Axioms
1. **Definition 1**: A property \(P\) is *positive* if it is necessarily positive, i.e., \(\Box P(x)\) implies something desirable or perfect.
– Formally: Let \(P\) be a property. \(P\) is positive if for all \(x\), \(P(x)\) is a “good” or “perfect” property.
2. **Definition 2**: An object \(x\) is *god-like* (denoted \(G(x)\)) if \(x\) possesses all positive properties.
\[
G(x) \iff \forall P (P \text{ is positive} \to P(x)).
\]
3. **Definition 3**: A property \(E\) is the *essence* of \(x\) if \(E(x)\) entails all properties of \(x\) and is necessarily true of \(x\).
\[
E \text{ is an essence of } x \iff E(x) \land \forall Q (Q(x) \to \Box \forall y (E(y) \to Q(y))).
\]
4. **Definition 4**: *Necessary existence* (\(NE(x)\)) holds if the essence of \(x\) implies that \(x\) exists in all possible worlds.
\[
NE(x) \iff \forall E (E \text{ is an essence of } x \to \Box \exists y E(y)).
\]

**Axioms**:
– **Axiom 1**: If \(P\) is a positive property, then its negation \(\neg P\) is not positive.
\[
P \text{ is positive} \to \neg (\neg P \text{ is positive}).
\]
– **Axiom 2**: Any property entailed by a positive property is positive.
\[
\forall P \forall Q [(P \text{ is positive} \land \Box \forall x (P(x) \to Q(x))) \to Q \text{ is positive}].
\]
– **Axiom 3**: The property of being god-like (\(G\)) is positive.
\[
G \text{ is positive}.
\]
– **Axiom 4**: If a property is positive, it is necessarily positive.
\[
P \text{ is positive} \to \Box (P \text{ is positive}).
\]
– **Axiom 5**: Necessary existence (\(NE\)) is a positive property.
\[
NE \text{ is positive}.
\]

**Modal Logic System**:
– Gödel uses a modal logic system akin to **S5**, where:
– \(\Box P \to P\) (what is necessary is true).
– \(\Diamond P \to \Box \Diamond P\) (if something is possible, it is necessarily possible).
– \(\Box P \to \Box \Box P\) (if something is necessary, it is necessarily necessary).

The Proof
The proof proceeds in a series of theorems to show that a god-like being exists necessarily.

**Theorem 1**: It is possible that a god-like being exists.
– By Axiom 3, \(G\) is a positive property.
– Assume no god-like being exists in any possible world, i.e., \(\neg \Diamond \exists x G(x)\).
– This implies \(\Box \forall x \neg G(x)\).
– By Definition 2, \(G(x) \iff \forall P (P \text{ is positive} \to P(x))\). If \(\neg G(x)\), then there exists some positive property \(P\) such that \(\neg P(x)\).
– By Axiom 1, if \(P\) is positive, \(\neg P\) is not positive. Thus, \(\neg G(x)\) implies some non-positive property is held.
– If \(\Box \forall x \neg G(x)\), then in all possible worlds, every \(x\) lacks some positive property. But this contradicts Axiom 4 (positive properties are necessarily positive) and the consistency of positive properties across worlds.
– Hence, the assumption \(\neg \Diamond \exists x G(x)\) leads to a contradiction.
– Therefore, \(\Diamond \exists x G(x)\).

\[
\Diamond \exists x G(x).
\]

**Theorem 2**: If a god-like being is possible, it exists in all possible worlds.
– Suppose \(\exists x G(x)\) in some possible world \(w\).
– By Definition 2, \(G(x)\) means \(x\) has all positive properties, including \(NE\) (by Axiom 5).
– By Definition 4, \(NE(x)\) means the essence of \(x\) implies \(\Box \exists y G(y)\).
– The essence of \(x\) includes \(G(x)\), since \(G(x)\) entails all positive properties of \(x\).
– Thus, \(G(x) \to NE(x) \to \Box \exists y G(y)\).
– In world \(w\), if \(\exists x G(x)\), then \(\Box \exists y G(y)\), meaning a god-like being exists in all possible worlds.
– In S5, \(\Diamond \Box P \to \Box P\). Since \(\Diamond \exists x G(x)\), and \(G(x) \to \Box \exists y G(y)\), we have:
\[
\Diamond \exists x G(x) \to \Box \exists x G(x).
\]

**Conclusion**:
– From Theorem 1, \(\Diamond \exists x G(x)\).
– From Theorem 2, \(\Diamond \exists x G(x) \to \Box \exists x G(x)\).
– Therefore, \(\Box \exists x G(x)\), i.e., a god-like being exists necessarily.
– Since \(\Box P \to P\), it follows that \(\exists x G(x)\).

\[
\exists x G(x).
\]

Thus, a god-like being, possessing all positive properties, exists.

Notes
– **Positive Properties**: Gödel leaves “positive” abstract, but it can be interpreted as properties contributing to perfection (e.g., omnipotence, omniscience, omnibenevolence).
– **Modal Logic**: The proof relies heavily on the S5 axiom \(\Diamond P \to \Box \Diamond P\), which some philosophers debate.
– **Criticisms**: Critics argue the proof assumes the coherence of “positive properties” and the acceptability of S5. Others question whether existence is a property.