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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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  • 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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  • Self-Reference Upfront: A Study of Self-Referential Gödel Numberings.Balthasar Grabmayr & Albert Visser - 2023 - Review of Symbolic Logic 16 (2):385-424.
    In this paper we examine various requirements on the formalisation choices under which self-reference can be adequately formalised in arithmetic. In particular, we study self-referential numberings, which immediately provide a strong notion of self-reference even for expressively weak languages. The results of this paper suggest that the question whether truly self-referential reasoning can be formalised in arithmetic is more sensitive to the underlying coding apparatus than usually believed. As a case study, we show how this sensitivity affects the formal study (...)
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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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