7,009 research outputs found
Wittgenstein’s ‘notorious paragraph’ about the Gödel Theorem
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
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 (), as
possible seeds of CMB anisotropies along with the standard inflationary
perturbations. With the Planck 2018 observations, we find
and 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
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
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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