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."
arXiv:2607. 10880v1 Announce Type: new Abstract: We extend, in Isabelle/HOL, the deep-and-shallow embedding methodology of our prior work from propositional to first-order modal logic (FML) with constant-domain Kripke semantics.
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.
arXiv:2606. 28841v1 Announce Type: cross Abstract: Large language models are increasingly capable of mathematical reasoning, but the proofs they generate are often unreliable and hard to verify.
arXiv:2607. 16372v1 Announce Type: cross Abstract: Interactive theorem proving (ITP) underpins program verification and formalized mathematics, but its manual effort limits scalability.
arXiv:2607. 12650v1 Announce Type: cross Abstract: Tool access alone does not make LLM empirical reasoning governable: accepted outputs need not descend from attested evidence, and accepted deductions need not hold up under formal scrutiny.
We propose ZX-Calculus (Knowledge Evolution Calculus), a conservative extension of Martin-Lof Dependent Type Theory (MLTT) integrating trace-indexed types, presheaf non-monotone semantics, and constructive AGM belief revision. A Coq mechanisation accompanies the paper (34 complete proofs; zero admits for the two central results).
arXiv:2607. 06341v1 Announce Type: cross Abstract: Formal verification offers the strongest guarantee of software correctness, but it does not scale: the proofs demanded by interactive theorem provers such as Coq require enormous expert effort.
arXiv:2608. 05420v1 Announce Type: cross Abstract: Large language models (LLMs) can generate text that resembles a mathematical proof, but resemblance does not establish correctness.
arXiv:2608. 15432v1 Announce Type: new Abstract: In formal verification, both the autoformalization of statements and automated proof search have been studied extensively.
arXiv:2607. 15629v1 Announce Type: cross Abstract: Topos causal models recast causal inference inside a topos: a causal world is a presheaf, an intervention is a characteristic map into the subobject classifier, and reasoning is carried out in the intuitionistic internal language.
Formal verification offers the strongest guarantee of software correctness, but it does not scale: the proofs demanded by interactive theorem provers such as Coq require enormous expert effort. Large language models (LLMs) promise to generate these proofs automatically, yet existing approaches wire a fixed, human-designed proof strategy into the system and constrain the model to follow it (retrieving premises and predicting tactics one step at a time, or splitting goals by divide-and-conquer), and still prove only a fraction of their target theorems.
arXiv:2606. 06468v1 Announce Type: new Abstract: We introduce Goedel-Architect, an agentic framework for formal theorem proving in Lean 4 centered on blueprint generation and refinement.