9 found
Sort by:
  1. Peter A. Cholak, Damir D. Dzhafarov & Jeffry L. Hirst (2012). On Mathias Generic Sets. In. In S. Barry Cooper (ed.), How the World Computes. 129--138.
    No categories
    Direct download (2 more)  
     
    My bibliography  
     
    Export citation  
  2. Damir D. Dzhafarov & Carl Mummert (2012). Reverse Mathematics and Properties of Finite Character. Annals of Pure and Applied Logic 163 (9):1243-1251.
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  3. David E. Diamondstone, Damir D. Dzhafarov & Robert I. Soare (2010). $\Pi^0_1$ Classes, Peano Arithmetic, Randomness, and Computable Domination. Notre Dame Journal of Formal Logic 51 (1):127-159.
    We present an overview of the topics in the title and of some of the key results pertaining to them. These have historically been topics of interest in computability theory and continue to be a rich source of problems and ideas. In particular, we draw attention to the links and connections between these topics and explore their significance to modern research in the field.
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  4. Damir D. Dzhafarov (2010). Stable Ramsey's Theorem and Measure. Notre Dame Journal of Formal Logic 52 (1):95-112.
    The stable Ramsey's theorem for pairs has been the subject of numerous investigations in mathematical logic. We introduce a weaker form of it by restricting from the class of all stable colorings to subclasses of it that are nonnull in a certain effective measure-theoretic sense. We show that the sets that can compute infinite homogeneous sets for nonnull many computable stable colorings and the sets that can compute infinite homogeneous sets for all computable stable colorings agree below $\emptyset'$ but not (...)
    Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  5. Damir D. Dzhafarov, Jeffry L. Hirst & Tamara J. Lakins (2010). Ramsey's Theorem for Trees: The Polarized Tree Theorem and Notions of Stability. [REVIEW] Archive for Mathematical Logic 49 (3):399-415.
    We formulate a polarized version of Ramsey’s theorem for trees. For those exponents greater than 2, both the reverse mathematics and the computability theory associated with this theorem parallel that of its linear analog. For pairs, the situation is more complex. In particular, there are many reasonable notions of stability in the tree setting, complicating the analysis of the related results.
    No categories
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  6. Ehtibar N. Dzhafarov & Damir D. Dzhafarov (2010). Sorites Without Vagueness I: Classificatory Sorites. Theoria 76 (1):4-24.
    An abstract mathematical theory is presented for a common variety of soritical arguments, treated here in terms of responses of a system, say, a biological organism, a gadget, or a set of normative linguistic rules, to stimuli. Any characteristic of the system's responses which supervenes on stimuli is called a stimulus effect upon the system. Classificatory sorites is about the identity of or difference between the effects of stimuli that differ 'only microscopically'. We formulate the classificatory sorites on arguably the (...)
    Direct download (7 more)  
     
    My bibliography  
     
    Export citation  
  7. Ehtibar N. Dzhafarov & Damir D. Dzhafarov (2010). Sorites Without Vagueness II: Comparative Sorites. Theoria 76 (1):25-53.
    We develop a mathematical theory for comparative sorites, considered in terms of a system mapping pairs of stimuli into a binary response characteristic whose values supervene on stimulus pairs and are interpretable as the complementary relations 'are the same' and 'are not the same' (overall or in some respect). Comparative sorites is about hypothetical sequences of stimuli in which every two successive elements are mapped into the relation 'are the same', while the pair comprised of the first and the last (...)
    Direct download (7 more)  
     
    My bibliography  
     
    Export citation  
  8. Damir D. Dzhafarov & Jeffry L. Hirst (2009). The Polarized Ramsey's Theorem. Archive for Mathematical Logic 48 (2):141-157.
    We study the effective and proof-theoretic content of the polarized Ramsey’s theorem, a variant of Ramsey’s theorem obtained by relaxing the definition of homogeneous set. Our investigation yields a new characterization of Ramsey’s theorem in all exponents, and produces several combinatorial principles which, modulo bounding for ${\Sigma^0_2}$ formulas, lie (possibly not strictly) between Ramsey’s theorem for pairs and the stable Ramsey’s theorem for pairs.
    No categories
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  9. Damir D. Dzhafarov & Carl G. Jockusch Jr (2009). Ramsey's Theorem and Cone Avoidance. Journal of Symbolic Logic 74 (2):557 - 578.
    It was shown by Cholak, Jockusch, and Slaman that every computable 2-coloring of pairs admits an infinite low₂ homogeneous set H. We answer a question of the same authors by showing that H may be chosen to satisfy in addition $C\,\not \leqslant _T \,H$ , where C is a given noncomputable set. This is shown by analyzing a new and simplified proof of Seetapun's cone avoidance theorem for Ramsey's theorem. We then extend the result to show that every computable 2-coloring (...)
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation