17 found
Order:
  1. The negative theology of absolute infinity: Cantor, mathematics, and humility.Rico Gutschmidt & Merlin Carl - 2024 - International Journal for Philosophy of Religion 95 (3):233-256.
    Cantor argued that absolute infinity is beyond mathematical comprehension. His arguments imply that the domain of mathematics cannot be grasped by mathematical means. We argue that this inability constitutes a foundational problem. For Cantor, however, the domain of mathematics does not belong to mathematics, but to theology. We thus discuss the theological significance of Cantor’s treatment of absolute infinity and show that it can be interpreted in terms of negative theology. Proceeding from this interpretation, we refer to the recent debate (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  2.  58
    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)  
     
    Export citation  
     
    Bookmark   8 citations  
  3.  30
    Ordinal Computability: An Introduction to Infinitary Machines.Merlin Carl - 2019 - Boston: 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.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  4.  96
    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 (4 more)  
     
    Export citation  
     
    Bookmark   7 citations  
  5.  29
    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  
  6.  25
    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  
  7. (1 other version)Formal and Natural Proof: A Phenomenological Approach.Merlin Carl - 2019 - In Stefania Centrone, Deborah Kant & Deniz Sarikaya (eds.), Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts. Springer Verlag.
    No categories
     
    Export citation  
     
    Bookmark   1 citation  
  8.  39
    The distribution of ITRM-recognizable reals.Merlin Carl - 2014 - Annals of Pure and Applied Logic 165 (9):1403-1417.
    Infinite Time Register Machines are a well-established machine model for infinitary computations. Their computational strength relative to oracles is understood, see e.g. , and . We consider the notion of recognizability, which was first formulated for Infinite Time Turing Machines in [6] and applied to ITRM 's in [3]. A real x is ITRM -recognizable iff there is an ITRM -program P such that PyPy stops with output 1 iff y=xy=x, and otherwise stops with output 0. In [3], it is (...))
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  9.  4
    Sociomathematical Norms and Automated Proof Checking in Mathematical Education: Reflections and Experiences.Merlin Carl - unknown
    According to a widely held view, mathematical proofs are essentially (indications of) formal derivations, and thus in principle mechanically checkable (this view is defended, for example, by Azzouni [3]). This should in particular hold for the kind of simple proof exercises typically given to students of mathematics learning to write proofs. If that is so, then automated proof checking should be an attractive option for math education at the undergraduate level. An opposing view would be that mathematical proofs are social (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  10.  26
    Realisability for infinitary intuitionistic set theory.Merlin Carl, Lorenzo Galeotti & Robert Passmann - 2023 - Annals of Pure and Applied Logic 174 (6):103259.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  11.  19
    Optimal results on recognizability for infinite time register machines.Merlin Carl - 2015 - Journal of Symbolic Logic 80 (4):1116-1130.
  12.  23
    Understanding mathematical texts: a hermeneutical approach.Merlin Carl - 2022 - Synthese 200 (6):1–31.
    The work done so far on the understanding of mathematical (proof) texts focuses mostly on logical and heuristical aspects; a proof text is considered to be understood when the reader is able to justify inferential steps occurring in it, to defend it against objections, to give an account of the “main ideas”, to transfer the proof idea to other contexts etc. (see, e.g., Avigad in The philosophy of mathematical practice, Oxford University Press, Oxford, 2008). In contrast, there is a rich (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  13.  38
    Taming Koepke's Zoo II: Register machines.Merlin Carl - 2022 - Annals of Pure and Applied Logic 173 (3):103041.
  14.  25
    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  
  15.  20
    Canonical Truth.Merlin Carl & Philipp Schlicht - 2022 - Axiomathes 32 (3):785-803.
    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  
  16.  11
    Correction to: Formal and Natural Proof: A Phenomenological Approach.Merlin Carl - 2019 - In Stefania Centrone, Deborah Kant & Deniz Sarikaya (eds.), Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts. Springer Verlag.
    The abstract of this chapter was initially published with error. The chapter has been updated with the corrected abstract as given below.
    No categories
    Direct download  
     
    Export citation  
     
    Bookmark  
  17.  26
    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