Abstract
In this paper I try to discuss the question of the truth-value of Gödel-type undecidable sentences in a framework which keeps into due account the idea that mathematical inquiry develops in a three-level framework: informal (or pre-formal) mathematics, (informal) theories, and formal theories. Moreover, it is to be stressed that no phase deletes the other ones; all of them, so to speak, live together.
I wish to thank my colleague Luca Bellotti for many helpful discussions on these topics.
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Notes
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We report here the italian text:
Ciò che viene accettato concordemente da tutti i matematici è il carattere specifico di tutti gli enunciati matematici, delle dimostrazioni matematiche. È assolutamente fuori discussione fra i matematici che cosa voglia dire che “un teorema è sensato”, ossia scritto in modo coerente alle regole interne della matematica e che “una dimostrazione è giusta” o “è sbagliata”. Su questo esiste una totale unanimità fra i matematici.
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For a thorough treatment of the question of self-reference one can see Halbach and Visser (2014).
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What I mean is the impossibility to have in PA the uniform assertion of its soundness. It is well known, on the other hand, that it is possible to build partial truth definitions which, given, say, an arithmetical sentence A ∈ Σ n, allow to formalize in PA the obvious intuitive implication from provability to truth with respect to sentences of that complexity.
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The question of (the sense in which one can speak about) the truth of G has been vastly debated: we limit to mention the very recent (Piazza and Pulcini 2013, 2015, 2016); each of them contains many pertinent references. For an interesting survey of the entire debate concerning the assessment of the (possible) truth-value of G, one can profitably see also (Pantsar 2009).
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Moriconi, E. (2018). Some Remarks on True Undecidable Sentences. In: Piazza, M., Pulcini, G. (eds) Truth, Existence and Explanation. Boston Studies in the Philosophy and History of Science, vol 334. Springer, Cham. https://doi.org/10.1007/978-3-319-93342-9_1
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