The Ontological Argument

Proofs of the Existence of God

• Zalta's first-order proof based on Anselm's argument
• Rushby's higher-order proof based on Anselm's argument
• Benzmüller and Woltzenlogel-Paleo's second-order modal proof
  based on Gödel's argument
  • Dana Scott typed up Gödel's handwritten presentation.
  • Gödel's axioms are contradictory, and Scott fixed that.
  • Gödel's and Scott's axioms have P ⇒ □P (modal collapse) as a theorem.
  • Benzmüller produced a simplified version that avoids modal collapse.

The Benzmüller and Woltzenlogel-Paleo Proof

• Formulae encoded in second-order modal logic
• Second-order modal logic translated to THF
• Incremental development using TPI
• Proofs discharged via SystemOnTPTP

See the press coverage, e.g., Der Spiegel