A Step Towards Absolute Versions of Metamathematical Results

Article Mathematics, Applied

SELF-REFERENCE UPFRONT: A STUDY OF SELF-REFERENTIAL GoDEL NUMBERINGS

Balthasar Grabmayr et al.

Summary: In this paper, we examine the various requirements for adequately formalizing self-reference in arithmetic and find that it depends on the underlying coding apparatus. Through a case study, the paper demonstrates how this sensitivity affects the formal study of certain principles of self-referential truth.

REVIEW OF SYMBOLIC LOGIC (2023)

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Article Logic

Varieties of Self-Reference in Metamathematics

Balthasar Grabmayr et al.

Summary: This paper explores the circumstances where diagonal sentences can be regarded as paradigmatic cases of self-reference. By introducing well-motivated constraints on the diagonal operator and the coding apparatus, we distinguish paradigmatic self-referential sentences, such as those obtained through Godel's diagonalization method, from accidental diagonal sentences. Specifically, we demonstrate the successful exclusion of refutable Henkin sentences constructed by Kreisel.

JOURNAL OF PHILOSOPHICAL LOGIC (2023)

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Article Mathematics, Applied

ON THE INVARIANCE OF GoDEL'S SECOND THEOREM WITH REGARD TO NUMBERINGS

Balthasar Grabmayr

Summary: This paper examines the dependency of Godel's Second Theorem on Godel numberings, introducing deviant numberings that lead to provable consistency predicates. However, the main result suggests that these counterexamples do not refute the prevalent interpretation of the theorem, as invariance is maintained once a natural class of admissible numberings is identified.

REVIEW OF SYMBOLIC LOGIC (2021)

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Article Mathematics

CURRENT RESEARCH ON GoDEL'S INCOMPLETENESS THEOREMS

Yong Cheng

Summary: The article provides an overview of current research on Godel's incompleteness theorems, focusing on the classification of different proofs, the limitations of the applicability of Godel's first incompleteness theorem, and the limitations of the applicability of Godel's second incompleteness theorem.

BULLETIN OF SYMBOLIC LOGIC (2021)

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Article Mathematics, Applied