Workshop: Computability and Provability
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| Workshop: Computability and Provability
Organised by Leon Horsten and Philip Welch Contact: Leon.Horsten@bristol.ac.uk Friday, 19 March 2010 Institute for Advanced Studies, Verdon Smith Room, University of Bristol Schedule: 9.30h -11.00h: Peter Koellner (Harvard): Gödel’s dichotomy 11.15h - 12.45h: Mark Sprevak (Cambridge): Real computation 12.45 – 14.00h: Lunch 14.00h -15.30h: Walter Dean (Warwick): Models and recursivity 16.00h – 17.30h: Leon Horsten (Bristol): Structure and computation (for this talk we will move to room 19WR G99A) Titles and abstracts: (1) Peter Koellner (Harvard) Gödel’s dichotomy There has been a great deal of discussion of concerning the connection between Gödel's theorems and the question of whether the mind can be mechanized. We now know (from the unpublished papers) that Gödel had a quite subtle position on this matter. His position is embodied in the following dichotomy: Either the human mind infinitely surpasses any Turing machine or there exist absolutely undecidable sentences. More recent thinkers – most notably, Lucas and Penrose -- have argued for a stronger conclusion, that is, they have argued for the first disjunct of Gödel's Dichotomy. In this talk I will discuss these and related issues. The three main points I shall defend are as follows: First, the proper setting in which to discuss these issues is that of Reinhardt's epistemic arithmetic. In this setting one can obtain definitive results. In particular, one can prove Gödel's Dichotomy and there are two fundamental theorems of Reinhardt and Carlson which bring out a subtle distinction and undermine the arguments of Lucas and Penrose. Second, Penrose's new argument has not been given a proper analysis. The proper analysis requires formalizing knowledge (or absolute provability) with an operator (not a predicate) and formalizing truth with an unrestricted truth predicate. This can be done and when it is done something interesting emerges. Third, there is a way of granting that there are absolutely undecidable statements (something that is inevitable in light of the incompleteness theorems) while undermining the significance of this fact. This last point will turn on a scenario for an open-ended sequence of axioms that settles CH and the other undecided statements of set theory. (2) Mark Sprevak (Cambridge) Real computational structure What does it mean for a physical system to implement a computation? Is the computational structure of a physical system a real property of that system, or only something that observers project onto it? This paper examines an argument by Putnam (1988) that appears to demonstrate that there are no non-trivial constraints on the computational structure of a physical system. I consider existing responses to Putnam's argument, and I argue that none of them work. I argue that a line of response is available that blocks Putnam's worry, and that yields real computational structure for physical systems, but the cost of this response is that it ties the notion of representation closer to the notion of computation than many would like. (3) Walter Dean (Warwick) Models and recursivity In this talk, I will examine the relevance of broadly computational considerations to securing the determinacy of mathematical truth and reference. I will focus in particular on arguments which attempt to employ Tennenbaum's Theorem to “rule out” non-standard models of first-order arithmetic (versions of which have been cited by Halbach and Horsten [2005], Dean [2002] and Quinon and Zdanowski(2007)). I will argue that such arguments turn on a number of equivocations. I will then consider whether results from complexity theory can be brought to bear on this issue. (4) Leon Horsten (Bristol) Structure and computation In this talk, I will investigate the question how we, in our arithmetical practice, succeed in singling out the natural number structure as our intended interpretation. I will argue that this is effected by a combination of what we assert about the natural number structure and our computational capacities. Directory: media-library media-library -> Federal emergency management agency fema rep-2, rev. 2 / June 1990 media-library -> Availability notice media-library -> Civil aviation directorate media-library -> Environmental Assessment Horizon Monitoring and Surveillance Facility Puerto Nuevo Port Complex, San Juan, Puerto Rico media-library -> Final hurricane script media-library -> Listing of Disaster Recovery Funding Resources media-library -> Nfip perspectives: History & Background Transcript Download 11.37 Kb. Share with your friends:
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