7,009 research outputs found

    Wittgenstein’s ‘notorious paragraph’ about the Gödel Theorem

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    In §8 of Remarks on the Foundations of Mathematics (RFM), Appendix 3 Wittgenstein imagines what conclusions would have to be drawn if the Gödel formula P or ¬P would be derivable in PM. In this case, he says, one has to conclude that the interpretation of P as “P is unprovable” must be given up. This “notorious paragraph” has heated up a debate on whether the point Wittgenstein has to make is one of “great philosophical interest” revealing “remarkable insight” in Gödel’s proof, as Floyd and Putnam suggest (Floyd (2000), Floyd (2001)), or whether this remark reveals Wittgenstein’s misunderstanding of Gödel’s proof as Rodych and Steiner argued for recently (Rodych (1999, 2002, 2003), Steiner (2001)). In the following the arguments of both interpretations will be sketched and some deficiencies will be identified. Afterwards a detailed reconstruction of Wittgenstein’s argument will be offered. It will be seen that Wittgenstein’s argumentation is meant to be a rejection of Gödel’s proof but that it cannot satisfy this pretension

    On the Detectability of CMB Anisotropies Induced by de Sitter-G\"odel-de Sitter Phase Transition

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    A geometrical phase transition in the very early Universe, from de Sitter to G\"odel and back to de Sitter (dGd) spacetimes, can explain the universal phenomenon of rotation of many large scale structures. This phase transition is shown to induce fluctuations on the matter and radiation fields with possibly observable traces. In this work we simulate the dGd-induced inhomogeneities and use their power spectrum, parametrized by the parameter pair (p1,p2p_1, p_2), as possible seeds of CMB anisotropies along with the standard inflationary perturbations. With the Planck 2018 observations, we find p1=0.0080.008+0.003p_1=0.008^{+0.003}_{-0.008} and p2=0.0020.002+0.001p_2= 0.002^{+0.001}_{-0.002} consistent with pure inflationary power spectrum and no hint for the dGd transition. Future large scale surveys can further tighten the constraints and probe the physics of the early Universe

    Consequences of a Goedel's misjudgment

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    The fundamental aim of the paper is to correct an harmful way to interpret a Goedel's erroneous remark at the Congress of Koenigsberg in 1930. Despite the Goedel's fault is rather venial, its misreading has produced and continues to produce dangerous fruits, as to apply the incompleteness Theorems to the full second-order Arithmetic and to deduce the semantic incompleteness of its language by these same Theorems. The first three paragraphs are introductory and serve to define the languages inherently semantic and its properties, to discuss the consequences of the expression order used in a language and some question about the semantic completeness: in particular is highlighted the fact that a non-formal theory may be semantically complete despite using a language semantically incomplete. Finally, an alternative interpretation of the Goedel's unfortunate comment is proposed. KEYWORDS: semantic completeness, syntactic incompleteness, categoricity, arithmetic, second-order languages, paradoxesComment: English version, 19 pages. Fixed and improved terminolog

    Predicativity and parametric polymorphism of Brouwerian implication

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    A common objection to the definition of intuitionistic implication in the Proof Interpretation is that it is impredicative. I discuss the history of that objection, argue that in Brouwer's writings predicativity of implication is ensured through parametric polymorphism of functions on species, and compare this construal with the alternative approaches to predicative implication of Goodman, Dummett, Prawitz, and Martin-L\"of.Comment: Added further references (Pistone, Poincar\'e, Tabatabai, Van Atten
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