15 found
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  1.  15
    Characterizations of Pretameness and the Ord-Cc.Peter Holy, Regula Krapf & Philipp Schlicht - 2018 - Annals of Pure and Applied Logic 169 (8):775-802.
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  2.  11
    The Exact Strength of the Class Forcing Theorem.Victoria Gitman, Joel David Hamkins, Peter Holy, Philipp Schlicht & Kameryn J. Williams - 2020 - Journal of Symbolic Logic 85 (3):869-905.
    The class forcing theorem, which asserts that every class forcing notion ${\mathbb {P}}$ admits a forcing relation $\Vdash _{\mathbb {P}}$, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory $\text {GBC}$ to the principle of elementary transfinite recursion $\text {ETR}_{\text {Ord}}$ for class recursions of length $\text {Ord}$. It is also equivalent to the existence of truth predicates for the (...)
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  3.  10
    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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  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.
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  5.  11
    Continuous Reducibility and Dimension of Metric Spaces.Philipp Schlicht - 2018 - Archive for Mathematical Logic 57 (3-4):329-359.
    If is a Polish metric space of dimension 0, then by Wadge’s lemma, no more than two Borel subsets of X are incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space of positive dimension, there are uncountably many Borel subsets of that are pairwise incomparable with respect to continuous reducibility. In general, the reducibility that is given by the collection of continuous functions on a topological space \\) is called the Wadge quasi-order (...)
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  6.  36
    A Minimal Prikry-Type Forcing for Singularizing a Measurable Cardinal.Peter Koepke, Karen Räsch & Philipp Schlicht - 2013 - Journal of Symbolic Logic 78 (1):85-100.
    Recently, Gitik, Kanovei and the first author proved that for a classical Prikry forcing extension the family of the intermediate models can be parametrized by $\mathscr{P}(\omega)/\mathrm{finite}$. By modifying the standard Prikry tree forcing we define a Prikry-type forcing which also singularizes a measurable cardinal but which is minimal, i.e., there are \emph{no} intermediate models properly between the ground model and the generic extension. The proof relies on combining the rigidity of the tree structure with indiscernibility arguments resulting from the normality (...)
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  7.  20
    Lipschitz and Uniformly Continuous Reducibilities on Ultrametric Polish Spaces.Philipp Schlicht & Motto Ros Luca - 2014 - In Dieter Spreen, Hannes Diener & Vasco Brattka (eds.), Logic, Computation, Hierarchies. De Gruyter. pp. 213-258.
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  8.  9
    Long Games and Σ-Projective Sets.Juan P. Aguilera, Sandra Müller & Philipp Schlicht - 2021 - Annals of Pure and Applied Logic 172 (4):102939.
    We prove a number of results on the determinacy of σ-projective sets of reals, i.e., those belonging to the smallest pointclass containing the open sets and closed under complements, countable unions, and projections. We first prove the equivalence between σ-projective determinacy and the determinacy of certain classes of games of variable length <ω^2 (Theorem 2.4). We then give an elementary proof of the determinacy of σ-projective sets from optimal large-cardinal hypotheses (Theorem 4.4). Finally, we show how to generalize the proof (...)
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  9.  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 (...)
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  10.  11
    Preserving Levels of Projective Determinacy by Tree Forcings.Fabiana Castiblanco & Philipp Schlicht - 2021 - Annals of Pure and Applied Logic 172 (4):102918.
    We prove that various classical tree forcings—for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing—preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes are added to thin projective transitive relations by these forcings.
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  11. Ideal Topologies in Higher Descriptive Set Theory.Peter Holy, Marlene Koelbing, Philipp Schlicht & Wolfgang Wohofsky - forthcoming - Annals of Pure and Applied Logic:103061.
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  12.  7
    Measurable Cardinals and Good Σ1-Wellorderings.Philipp Lücke & Philipp Schlicht - 2018 - Mathematical Logic Quarterly 64 (3):207-217.
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  13.  4
    Coarse Groups, and the Isomorphism Problem for Oligomorphic Groups.André Nies, Philipp Schlicht & Katrin Tent - forthcoming - Journal of Mathematical Logic.
    Let S∞ denote the topological group of permutations of the natural numbers. A closed subgroup G of S∞ is called oligomorphic if for each n, its natural action on n-tuples of natural numbers has onl...
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  14.  42
    Automata on Ordinals and Automaticity of Linear Orders.Philipp Schlicht & Frank Stephan - 2013 - Annals of Pure and Applied Logic 164 (5):523-527.
    We investigate structures recognizable by finite state automata with an input tape of length a limit ordinal. At limits, the set of states which appear unboundedly often before the limit are mapped to a limit state. We describe a method for proving non-automaticity and apply this to determine the optimal bounds for the ranks of linear orders recognized by such automata.
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  15.  10
    Thin Equivalence Relations in Scaled Pointclasses.Ralf Schindler & Philipp Schlicht - 2011 - Mathematical Logic Quarterly 57 (6):615-620.
    For ordinals α beginning a Σ1 gap in equation image, where equation image is closed under number quantification, we give an inner model-theoretic proof that every thin equation image equivalence relation is equation image in a real parameter from the hypothesis equation image.
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