10 found
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  1.  10
    Ordinal Arithmetic and $\sigma_{1}$ -Elementarity.Timothy J. Carlson - 1999 - Archive for Mathematical Logic 38 (7):449-460.
    We will introduce a partial ordering $\preceq_1$ on the class of ordinals which will serve as a foundation for an approach to ordinal notations for formal systems of set theory and second-order arithmetic. In this paper we use $\preceq_1$ to provide a new characterization of the ubiquitous ordinal $\epsilon _{0}$.
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  2.  8
    Ordinal Arithmetic and [Mathematical Formula]-Elementarity.Timothy J. Carlson - 1999 - Archive for Mathematical Logic 38 (7):449-460.
  3.  6
    Patterns of Resemblance of Order 2.Timothy J. Carlson - 2009 - Annals of Pure and Applied Logic 158 (1-2):90-124.
    We will investigate patterns of resemblance of order 2 over a family of arithmetic structures on the ordinals. In particular, we will show that they determine a computable well ordering under appropriate assumptions.
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  4.  19
    Normal Forms for Elementary Patterns.Timothy J. Carlson & Gunnar Wilken - 2012 - Journal of Symbolic Logic 77 (1):174-194.
    A notation for an ordinal using patterns of resemblance is based on choosing an isominimal set of ordinals containing the given ordinal. There are many choices for this set meaning that notations are far from unique. We establish that among all such isominimal sets there is one which is smallest under inclusion thus providing an appropriate notion of normal form notation in this context. In addition, we calculate the elements of this isominimal set using standard notations based on collapsing functions. (...)
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  5.  5
    Tracking Chains of Σ 2 -Elementarity.Timothy J. Carlson & Gunnar Wilken - 2012 - Annals of Pure and Applied Logic 163 (1):23-67.
  6. Ranked Partial Structures.Timothy J. Carlson - 2003 - Journal of Symbolic Logic 68 (4):1109-1144.
    The theory of ranked partial structures allows a reinterpretation of several of the standard results of model theory and first-order logic and is intended to provide a proof-theoretic method which allows for the intuitions of model theory. A version of the downward Löwenheim-Skolem theorem is central to our development. In this paper we will present the basic theory of ranked partial structures and their logic including an appropriate version of the completeness theorem.
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  7.  9
    Michael Rathjen. Recent Advances in Ordinal Analysis: -CA and Related Systems. The Bulletin of Symbolic Logic, Vol. 1 , Pp. 468–485. [REVIEW]Timothy J. Carlson - 2000 - Bulletin of Symbolic Logic 6 (3):357-358.
  8.  16
    On the Conservativity of the Axiom of Choice Over Set Theory.Timothy J. Carlson - 2011 - Archive for Mathematical Logic 50 (7-8):777-790.
    We show that for various set theories T including ZF, T + AC is conservative over T for sentences of the form ${\forall x \exists ! y}$ A(x, y) where A(x, y) is a Δ0 formula.
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  9.  17
    Knowledge, Machines, and the Consistency of Reinhardt's Strong Mechanistic Thesis.Timothy J. Carlson - 2000 - Annals of Pure and Applied Logic 105 (1--3):51--82.
    Reinhardt 's strong mechanistic thesis, a formalization of “I know I am a Turing machine”, is shown to be consistent with Epistemic Arithmetic.
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  10.  5
    Review: Michael Rathjen, Recent Advances in Ordinal Analysis: $Prod_{2}^{1}$-$CA$ and Related Systems. [REVIEW]Timothy J. Carlson - 2000 - Bulletin of Symbolic Logic 6 (3):357-358.