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  1. Essential hereditary undecidability.Albert Visser - forthcoming - Archive for Mathematical Logic:1-34.
    In this paper we study essential hereditary undecidability. Theories with this property are a convenient tool to prove undecidability of other theories. The paper develops the basic facts concerning essentially hereditary undecidability and provides salient examples, like a construction of essentially hereditarily undecidable theories due to Hanf and an example of a rather natural essentially hereditarily undecidable theory strictly below. We discuss the (non-)interaction of essential hereditary undecidability with recursive boolean isomorphism. We develop a reduction relation essential tolerance, or, in (...)
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  • Tarski’s Undefinability Theorem and the Diagonal Lemma.Saeed Salehi - 2022 - Logic Journal of the IGPL 30 (3):489-498.
    We prove the equivalence of the semantic version of Tarski’s theorem on the undefinability of truth with the semantic version of the diagonal lemma and also show the equivalence of a syntactic version of Tarski’s undefinability theorem with a weak syntactic diagonal lemma. We outline two seemingly diagonal-free proofs for these theorems from the literature and show that the syntactic version of Tarski’s theorem can deliver Gödel–Rosser’s incompleteness theorem.
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  • HYPER-REF: A General Model of Reference for First-Order Logic and First-Order Arithmetic.Pablo Rivas-Robledo - 2022 - Kriterion – Journal of Philosophy 36 (2):179-205.
    In this article I present HYPER-REF, a model to determine the referent of any given expression in First-Order Logic. I also explain how this model can be used to determine the referent of a first-order theory such as First-Order Arithmetic. By reference or referent I mean the non-empty set of objects that the syntactical terms of a well-formed formula pick out given a particular interpretation of the language. To do so, I will first draw on previous work to make explicit (...)
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  • Gödel’s Theorem and Direct Self-Reference.Saul A. Kripke - 2023 - Review of Symbolic Logic 16 (2):650-654.
    In his paper on the incompleteness theorems, Gödel seemed to say that a direct way of constructing a formula that says of itself that it is unprovable might involve a faulty circularity. In this note, it is proved that ‘direct’ self-reference can actually be used to prove his result.
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  • Varieties of Self-Reference in Metamathematics.Balthasar Grabmayr, Volker Halbach & Lingyuan Ye - 2023 - Journal of Philosophical Logic 52 (4):1005-1052.
    This paper investigates the conditions under which diagonal sentences can be taken to constitute paradigmatic cases of self-reference. We put forward well-motivated constraints on the diagonal operator and the coding apparatus which separate paradigmatic self-referential sentences, for instance obtained via Gödel’s diagonalization method, from accidental diagonal sentences. In particular, we show that these constraints successfully exclude refutable Henkin sentences, as constructed by Kreisel.
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  • A Step Towards Absolute Versions of Metamathematical Results.Balthasar Grabmayr - 2024 - Journal of Philosophical Logic 53 (1):247-291.
    There is a well-known gap between metamathematical theorems and their philosophical interpretations. Take Tarski’s Theorem. According to its prevalent interpretation, the collection of all arithmetical truths is not arithmetically definable. However, the underlying metamathematical theorem merely establishes the arithmetical undefinability of a set of specific Gödel codes of certain artefactual entities, such as infix strings, which are true in the standard model. That is, as opposed to its philosophical reading, the metamathematical theorem is formulated (and proved) relative to a specific (...)
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