17 found
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  1.  29
    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 (...)
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  2.  14
    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.
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  3.  71
    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 (...)
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  4.  20
    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.
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  5.  14
    The negative theology of absolute infinity: Cantor, mathematics, and humility.Rico Gutschmidt & Merlin Carl - forthcoming - International Journal for Philosophy of Religion:1-24.
    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 (...)
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  6.  14
    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 (...)
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  7. Formal and Natural Proof: A Phenomenological Approach.Merlin Carl - 2019 - In Deniz Sarikaya, Deborah Kant & Stefania Centrone (eds.), Reflections on the Foundations of Mathematics. Springer Verlag.
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  8.  6
    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. pp. 315-343.
    In this section, we apply the notions obtained above to a famous historical example of a false proof. Our goal is to demonstrate that this proof shows a sufficient degree of distinctiveness for a formalization in a Naproche-like system and hence that automatic checking could indeed have contributed in this case to the development of mathematics. This example further demonstrates that even incomplete distinctivication can be sufficient for automatic checking and that actual mistakes may occur already in the margin between (...)
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  9.  14
    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 (...))
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  10.  5
    Optimal results on recognizability for infinite time register machines.Merlin Carl - 2015 - Journal of Symbolic Logic 80 (4):1116-1130.
  11.  10
    Realisability for infinitary intuitionistic set theory.Merlin Carl, Lorenzo Galeotti & Robert Passmann - 2023 - Annals of Pure and Applied Logic 174 (6):103259.
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  12.  16
    Decision Times of Infinite Computations.Merlin Carl, Philipp Schlicht & Philip Welch - 2022 - Notre Dame Journal of Formal Logic 63 (2).
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  13.  8
    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 (...)
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  14.  1
    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.
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  15.  14
    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 (...)
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  16.  20
    Taming Koepke's Zoo II: Register machines.Merlin Carl - 2022 - Annals of Pure and Applied Logic 173 (3):103041.
  17.  6
    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 (...)
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