arXiv AI By Christoph Benzm\"uller

Proofs Without Nominals: G\"odel's Ontological Argument, its Shallow Embedding, and the Open Questions of the Monatshefte Notes

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arXiv AI
Sep 24

G\"odel's and Scott's Variants of the Ontological Argument in Lean 4

This paper reports a full, structure‑preserving port of the Isabelle/HOL dataset on G"odel’s and Scott’s modal ontological arguments to Lean 4. The port consists of 30 Lean 4 modules that mirror the original theories in section structure, declaration order, and naming, and a comparison tool confirms that all 548 statements are identical. The Lean 4 development reproduces every proof from the Isabelle/HOL version—including the inconsistency of G"odel’s 1970 axioms, the repaired variants, Scott’s variant, modal collapse, monotheism, and the ultrafilter property—while also proving five previously unproven statements and documenting the remaining 45 as unresolved. "whyItMatters":"The work demonstrates that Lean 4 can faithfully replicate a complex formal development from Isabelle/HOL, providing a new, lean‑based foundation for further exploration of modal ontological arguments."

By Christoph Benzm\"uller
arXiv AI
Sep 10

Monadic Second-Order Logic in HOL: Deep and Shallow with Automated Faithfulness (Extended Preprint)

The paper presents three Isabelle/HOL embeddings of monadic second‑order logic (MSO): a deep embedding, a maximal‑shallow embedding, and a minimal‑shallow embedding that collapses formulas to bool. It introduces a two‑sorted substitution system that ensures capture‑avoiding substitution and proves the faithfulness of all embeddings. A fully mechanised two‑sorted downward Löwenheim‑Skolem theorem is established, showing that the minimal embedding recovers deep validity relative to countable assignments and aligns with both the general (Henkin‑style) and standard readings of MSO, while also demonstrating differences in classical MSO properties across the embeddings.

By Christoph Benzmueller, Daniel Kirchner