29,254 research outputs found
G\"odel Incompleteness and the Black Hole Information Paradox
Semiclassical reasoning suggests that the process by which an object
collapses into a black hole and then evaporates by emitting Hawking radiation
may destroy information, a problem often referred to as the black hole
information paradox. Further, there seems to be no unique prediction of where
the information about the collapsing body is localized. We propose that the
latter aspect of the paradox may be a manifestation of an inconsistent
self-reference in the semiclassical theory of black hole evolution. This
suggests the inadequacy of the semiclassical approach or, at worst, that
standard quantum mechanics and general relavity are fundamentally incompatible.
One option for the resolution for the paradox in the localization is to
identify the G\"odel-like incompleteness that corresponds to an imposition of
consistency, and introduce possibly new physics that supplies this
incompleteness. Another option is to modify the theory in such a way as to
prohibit self-reference. We discuss various possible scenarios to implement
these options, including eternally collapsing objects, black hole remnants,
black hole final states, and simple variants of semiclassical quantum gravity.Comment: 14 pages, 2 figures; revised according to journal requirement
The Surprise Examination Paradox and the Second Incompleteness Theorem
We give a new proof for Godel's second incompleteness theorem, based on
Kolmogorov complexity, Chaitin's incompleteness theorem, and an argument that
resembles the surprise examination paradox. We then go the other way around and
suggest that the second incompleteness theorem gives a possible resolution of
the surprise examination paradox. Roughly speaking, we argue that the flaw in
the derivation of the paradox is that it contains a hidden assumption that one
can prove the consistency of the mathematical theory in which the derivation is
done; which is impossible by the second incompleteness theorem.Comment: 8 page
Kriesel and Wittgenstein
Georg Kreisel (15 September 1923 - 1 March 2015) was a formidable mathematical
logician during a formative period when the subject was becoming
a sophisticated field at the crossing of mathematics and logic. Both with his
technical sophistication for his time and his dialectical engagement with mandates,
aspirations and goals, he inspired wide-ranging investigation in the metamathematics
of constructivity, proof theory and generalized recursion theory.
Kreisel's mathematics and interactions with colleagues and students have been
memorably described in Kreiseliana ([Odifreddi, 1996]). At a different level of
interpersonal conceptual interaction, Kreisel during his life time had extended
engagement with two celebrated logicians, the mathematical Kurt Gödel and
the philosophical Ludwig Wittgenstein. About Gödel, with modern mathematical
logic palpably emanating from his work, Kreisel has reflected and written
over a wide mathematical landscape. About Wittgenstein on the other hand,
with an early personal connection established Kreisel would return as if with
an anxiety of influence to their ways of thinking about logic and mathematics,
ever in a sort of dialectic interplay. In what follows we draw this out through
his published essaysâand one letterâboth to elicit aspects of influence in his
own terms and to set out a picture of Kreisel's evolving thinking about logic
and mathematics in comparative relief.Accepted manuscrip
Wittgensteinâs ânotorious paragraphâ 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 ÂŹP would be derivable in PM. In this case, he says, one
has to conclude that the interpretation of P as âP is
unprovableâ must be given up. This ânotorious paragraphâ
has heated up a debate on whether the point Wittgenstein
has to make is one of âgreat philosophical interestâ
revealing âremarkable insightâ in Gödelâs proof, as Floyd
and Putnam suggest (Floyd (2000), Floyd (2001)), or
whether this remark reveals Wittgensteinâs
misunderstanding of Gödelâs 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
Wittgensteinâs argument will be offered. It will be seen that
Wittgensteinâs argumentation is meant to be a rejection of
Gödelâs proof but that it cannot satisfy this pretension
On Incomplete XML Documents with Integrity Constraints
Abstract. We consider incomplete specifications of XML documents in the presence of schema information and integrity constraints. We show that integrity constraints such as keys and foreign keys affect consistency of such specifications. We prove that the consistency problem for incomplete specifications with keys and foreign keys can always be solved in NP. We then show a dichotomy result, classifying the complexity of the problem as NP-complete or PTIME, depending on the precise set of features used in incomplete descriptions.
Review of 'The Outer Limits of Reason' by Noson Yanofsky 403p (2013) (review revised 2019)
I give a detailed review of 'The Outer Limits of Reason' by Noson Yanofsky from a unified perspective of Wittgenstein and evolutionary psychology. I indicate that the difficulty with such issues as paradox in language and math, incompleteness, undecidability, computability, the brain and the universe as computers etc., all arise from the failure to look carefully at our use of language in the appropriate context and hence the failure to separate issues of scientific fact from issues of how language works. I discuss Wittgenstein's views on incompleteness, paraconsistency and undecidability and the work of Wolpert on the limits to computation. To sum it up: The Universe According to Brooklyn---Good Science, Not So Good Philosophy.
Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book âThe Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searleâ 2nd ed (2019). Those interested in more of my writings may see âTalking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019) and Suicidal Utopian Delusions in the 21st Century 4th ed (2019
Relevant Arithmetic and Mathematical Pluralism
In The Consistency of Arithmetic and elsewhere, Meyer claims to ârepealâ Goedelâs second incompleteness theorem. In this paper, I review his argument, and then consider two ways of understanding it: from the perspective of mathematical pluralism and monism, respectively. Is relevant arithmetic just another legitimate practice among many, or is it a rival of its classical counterpartâa corrective to Goedel, setting us back on the path to the (One) True Arithmetic? To help answer, I sketch a few worked examples from relevant mathematics, to see what a non-classical (re)formulation of mathematics might look like in practice. I conclude that, while it is unlikely that relevant arithmetic describes past and present mathematical practice, and so might be most acceptable as a pluralist enterprise, it may yet prescribe a more monistic future venture
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