10 found
Order:
  1.  8
    Ordinal Computability: An Introduction to Infinitary Machines.Merlin Carl - 2019 - De Gruyter.
    Ordinal Computability discusses models of computation obtained by generalizing classical models, such as Turing machines or register machines, to transfinite working time and space. In particular, recognizability, randomness, and applications to other areas of mathematics, including set theory and model theory, are covered.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  2.  14
    How to Frame Understanding in Mathematics: A Case Study Using Extremal Proofs.Merlin Carl, Marcos Cramer, Bernhard Fisseni, Deniz Sarikaya & Bernhard Schröder - 2021 - Axiomathes 31 (5):649-676.
    The frame concept from linguistics, cognitive science and artificial intelligence is a theoretical tool to model how explicitly given information is combined with expectations deriving from background knowledge. In this paper, we show how the frame concept can be fruitfully applied to analyze the notion of mathematical understanding. Our analysis additionally integrates insights from the hermeneutic tradition of philosophy as well as Schmid’s ideal genetic model of narrative constitution. We illustrate the practical applicability of our theoretical analysis through a case (...)
    Direct download (3 more)  
    Translate
     
     
    Export citation  
     
    Bookmark   2 citations  
  3.  70
    The Basic Theory of Infinite Time Register Machines.Merlin Carl, Tim Fischbach, Peter Koepke, Russell Miller, Miriam Nasfi & Gregor Weckbecker - 2010 - Archive for Mathematical Logic 49 (2):249-273.
    Infinite time register machines (ITRMs) are register machines which act on natural numbers and which are allowed to run for arbitrarily many ordinal steps. Successor steps are determined by standard register machine commands. At limit times register contents are defined by appropriate limit operations. In this paper, we examine the ITRMs introduced by the third and fourth author (Koepke and Miller in Logic and Theory of Algorithms LNCS, pp. 306–315, 2008), where a register content at a limit time is set (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   7 citations  
  4.  16
    Recognizable Sets and Woodin Cardinals: Computation Beyond the Constructible Universe.Merlin Carl, Philipp Schlicht & Philip Welch - 2018 - Annals of Pure and Applied Logic 169 (4):312-332.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  5.  11
    Infinite Computations with Random Oracles.Merlin Carl & Philipp Schlicht - 2017 - Notre Dame Journal of Formal Logic 58 (2):249-270.
    We consider the following problem for various infinite-time machines. If a real is computable relative to a large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it already computable? We show that the answer is independent of ZFC for ordinal Turing machines with and without ordinal parameters and give a positive answer for most other machines. For instance, we consider infinite-time Turing machines, unresetting and (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  6.  4
    Decision Times of Infinite Computations.Merlin Carl, Philipp Schlicht & Philip Welch - 2022 - Notre Dame Journal of Formal Logic 63 (2).
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  7.  3
    Canonical Truth.Merlin Carl & Philipp Schlicht - forthcoming - Axiomathes:1-19.
    We introduce and study some variants of a notion of canonical set theoretical truth. By this, we mean truth in a transitive proper class model M of ZFC that is uniquely characterized by some $$\in$$ ∈ -formula. We show that there are interesting statements that hold in all such models, but do not follow from ZFC, such as the ground model axiom and the nonexistence of measurable cardinals. We also study a related concept in which we only require M to (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  8.  14
    Taming Koepke's Zoo II: Register Machines.Merlin Carl - 2022 - Annals of Pure and Applied Logic 173 (3):103041.
  9.  4
    Optimal Results on Recognizability for Infinite Time Register Machines.Merlin Carl - 2015 - Journal of Symbolic Logic 80 (4):1116-1130.
  10.  10
    Randomness Via Infinite Computation and Effective Descriptive Set Theory.Merlin Carl & Philipp Schlicht - 2018 - Journal of Symbolic Logic 83 (2):766-789.
    We study randomness beyond${\rm{\Pi }}_1^1$-randomness and its Martin-Löf type variant, which was introduced in [16] and further studied in [3]. Here we focus on a class strictly between${\rm{\Pi }}_1^1$and${\rm{\Sigma }}_2^1$that is given by the infinite time Turing machines introduced by Hamkins and Kidder. The main results show that the randomness notions associated with this class have several desirable properties, which resemble those of classical random notions such as Martin-Löf randomness and randomness notions defined via effective descriptive set theory such as${\rm{\Pi (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark