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Philip Welch [30]Philip D. Welch [11]
  1.  27
    Godel's Disjunction: The Scope and Limits of Mathematical Knowledge.Leon Horsten & Philip Welch (eds.) - 2016 - Oxford, England: Oxford University Press UK.
    The logician Kurt Godel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is equivalent to a Turing machine (i.e., a computer), or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that intend to show that the (...)
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  2. Reflecting on Absolute Infinity.Philip Welch & Leon Horsten - 2016 - Journal of Philosophy 113 (2):89-111.
    This article is concerned with reflection principles in the context of Cantor’s conception of the set-theoretic universe. We argue that within such a conception reflection principles can be formulated that confer intrinsic plausibility to strong axioms of infinity.
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  3. Possible-worlds semantics for modal notions conceived as predicates.Volker Halbach, Hannes Leitgeb & Philip Welch - 2003 - Journal of Philosophical Logic 32 (2):179-223.
    If □ is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where □ is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of (...)
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  4. Revision Revisited.Leon Horsten, Graham E. Leigh, Hannes Leitgeb & Philip Welch - 2012 - Review of Symbolic Logic 5 (4):642-664.
    This article explores ways in which the Revision Theory of Truth can be expressed in the object language. In particular, we investigate the extent to which semantic deficiency, stable truth, and nearly stable truth can be so expressed, and we study different axiomatic systems for the Revision Theory of Truth.
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  5.  16
    Closed and unbounded classes and the härtig quantifier model.Philip D. Welch - 2022 - Journal of Symbolic Logic 87 (2):564-584.
    We show that assuming modest large cardinals, there is a definable class of ordinals, closed and unbounded beneath every uncountable cardinal, so that for any closed and unbounded subclasses $P, Q, {\langle L[P],\in,P \rangle }$ and ${\langle L[Q],\in,Q \rangle }$ possess the same reals, satisfy the Generalised Continuum Hypothesis, and moreover are elementarily equivalent. Examples of such P are Card, the class of uncountable cardinals, I the uniform indiscernibles, or for any n the class $C^{n}{=_{{\operatorname {df}}}}\{ \lambda \, | \, (...)
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  6. Proving Theorems from Reflection.Philip Welch - 2019 - In Deniz Sarikaya, Deborah Kant & Stefania Centrone (eds.), Reflections on the Foundations of Mathematics. Springer Verlag.
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  7.  82
    On the Consistency Strength of the Inner Model Hypothesis.Sy-David Friedman, Philip Welch & W. Hugh Woodin - 2008 - Journal of Symbolic Logic 73 (2):391 - 400.
  8. The Undecidability of Propositional Adaptive Logic.Leon Horsten & Philip Welch - 2007 - Synthese 158 (1):41-60.
    We investigate and classify the notion of final derivability of two basic inconsistency-adaptive logics. Specifically, the maximal complexity of the set of final consequences of decidable sets of premises formulated in the language of propositional logic is described. Our results show that taking the consequences of a decidable propositional theory is a complicated operation. The set of final consequences according to either the Reliability Calculus or the Minimal Abnormality Calculus of a decidable propositional premise set is in general undecidable, and (...)
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  9.  88
    Set-theoretic absoluteness and the revision theory of truth.Benedikt Löwe & Philip D. Welch - 2001 - Studia Logica 68 (1):21-41.
    We describe the solution of the Limit Rule Problem of Revision Theory and discuss the philosophical consequences of the fact that the truth set of Revision Theory is a complete 1/2 set.
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  10.  13
    Proving Theorems from Reflection.Philip Welch - 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. 79-97.
    We review some fundamental questions concerning the real line of mathematical analysis, which, like the Continuum Hypothesis, are also independent of the axioms of set theory, but are of a less ‘problematic’ nature, as they can be solved by adopting the right axiomatic framework. We contend that any foundations for mathematics should be able to simply formulate such questions as well as to raise at least the theoretical hope for their resolution.The usual procedure in set theory is to add so-called (...)
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  11.  26
    Some observations on truth hierarchies: A correction.Philip D. Welch - 2020 - Review of Symbolic Logic 13 (4):857-860.
    A correction is needed to our paper: to the definition contained within the statement of Lemma 1.5 and thus arguments around it in §3.
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  12.  25
    The undecidability of propositional adaptive logic.Leon Horsten & Philip Welch - 2009 - Synthese 169 (1):217-218.
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  13.  22
    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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  14.  56
    Comparing inductive and circular definitions: Parameters, complexity and games.Kai-Uwe Küdhnberger, Benedikt Löwe, Michael Möllerfeld & Philip Welch - 2005 - Studia Logica 81 (1):79 - 98.
    Gupta-Belnap-style circular definitions use all real numbers as possible starting points of revision sequences. In that sense they are boldface definitions. We discuss lightface versions of circular definitions and boldface versions of inductive definitions.
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  15.  33
    On the possibility, or otherwise, of hypercomputation.Philip D. Welch - 2004 - British Journal for the Philosophy of Science 55 (4):739-746.
    We claim that a recent article of P. Cotogno ([2003]) in this journal is based on an incorrect argument concerning the non-computability of diagonal functions. The point is that whilst diagonal functions are not computable by any function of the class over which they diagonalise, there is no ?logical incomputability? in their being computed over a wider class. Hence this ?logical incomputability? regrettably cannot be used in his argument that no hypercomputation can compute the Halting problem. This seems to lead (...)
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  16.  18
    Comparing Inductive and Circular Definitions: Parameters, Complexity and Games.Philip Welch, Kai–Uwe Kühnberger, Benedikt Löwe & Michael Möllerfeld - 2005 - Studia Logica 81 (1):79-98.
    Gupta-Belnap-style circular definitions use all real numbers as possible starting points of revision sequences. In that sense they are boldface definitions. We discuss lightface versions of circular definitions and boldface versions of inductive definitions.
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  17. Field on revenge.Agustin Rayo & Philip Welch - 2007 - In J. C. Beall (ed.), Revenge of the Liar: New Essays on the Paradox. Oxford University Press.
  18.  19
    Discrete transfinite computation models.Philip D. Welch - 2011 - In S. B. Cooper & Andrea Sorbi (eds.), Computability in Context: Computation and Logic in the Real World. World Scientific. pp. 375--414.
  19.  16
    Games and Ramsey-like cardinals.Dan Saattrup Nielsen & Philip Welch - 2019 - Journal of Symbolic Logic 84 (1):408-437.
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  20.  11
    Bounded Martin's Maximum, weak Erdӧs cardinals, and ψ Ac.David Asperó & Philip D. Welch - 2002 - Journal of Symbolic Logic 67 (3):1141-1152.
  21.  7
    When cardinals determine the power set: inner models and Härtig quantifier logic.Jouko Väänänen & Philip D. Welch - forthcoming - Mathematical Logic Quarterly.
    We show that the predicate “x is the power set of y” is ‐definable, if V = L[E] is an extender model constructed from a coherent sequences of extenders, provided that there is no inner model with a Woodin cardinal. Here is a predicate true of just the infinite cardinals. From this we conclude: the validities of second order logic are reducible to, the set of validities of the Härtig quantifier logic. Further we show that if no L[E] model has (...)
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  22.  41
    Bounded Martin's maximum, weak Erdős cardinals, and $\psi\sb {AC}$.David Asperó & Philip Welch - 2002 - Journal of Symbolic Logic 67 (3):1141-1152.
  23.  17
    Decision Times of Infinite Computations.Merlin Carl, Philipp Schlicht & Philip Welch - 2022 - Notre Dame Journal of Formal Logic 63 (2).
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  24.  26
    Bounded Martin's Maximum, Weak [image] Cardinals, and [image].David Asperó & Philip D. Welch - 2002 - Journal of Symbolic Logic 67 (3):1141 - 1152.
    We prove that a form of the $Erd\H{o}s$ property (consistent with $V = L\lbrack H_{\omega_2}\rbrack$ and strictly weaker than the Weak Chang's Conjecture at ω1), together with Bounded Martin's Maximum implies that Woodin's principle $\psi_{AC}$ holds, and therefore 2ℵ0 = ℵ2. We also prove that $\psi_{AC}$ implies that every function $f: \omega_1 \rightarrow \omega_1$ is bounded by some canonical function on a club and use this to produce a model of the Bounded Semiproper Forcing Axiom in which Bounded Martin's Maximum (...)
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  25.  7
    Asymmetric Cut and Choose Games.Christopher Henney-Turner, Peter Holy, Philipp Schlicht & Philip Welch - forthcoming - Bulletin of Symbolic Logic:1-31.
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  26.  19
    Local club condensation and l-likeness.Peter Holy, Philip Welch & Liuzhen Wu - 2015 - Journal of Symbolic Logic 80 (4):1361-1378.
  27.  38
    Erratum: The Undecidability of Propositional Adaptive Logic.Leon Horsten & Philip Welch - 2009 - Synthese 169 (1):217 - 218.
  28.  19
    Global square and mutual stationarity at the ℵn.Peter Koepke & Philip D. Welch - 2011 - Annals of Pure and Applied Logic 162 (10):787-806.
    We give the proof of a theorem of Jensen and Zeman on the existence of a global □ sequence in the Core Model below a measurable cardinal κ of Mitchell order ) equal to κ++, and use it to prove the following theorem on mutual stationarity at n.Let ω1 denote the first uncountable cardinal of V and set to be the class of ordinals of cofinality ω1.TheoremIf every sequence n m. In particular, there is such a model in which for (...)
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  29.  34
    Comparing incomparable Kleene degrees.Philip Welch - 1985 - Journal of Symbolic Logic 50 (1):55-58.
  30.  13
    Minimality in the ▵13-degrees.Philip Welch - 1987 - Journal of Symbolic Logic 52 (4):908 - 915.
    We show in ZFC, assuming all reals have sharps, that a countable collection of ▵ 1 3 -degrees without a minimal upper bound implies the existence of inner models with measurable cardinals.
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  31.  3
    Minimality in the $triangle^1_3$-Degrees.Philip Welch - 1987 - Journal of Symbolic Logic 52 (4):908-915.
    We show in ZFC, assuming all reals have sharps, that a countable collection of $\triangle^1_3$-degrees without a minimal upper bound implies the existence of inner models with measurable cardinals.
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  32.  1
    Minimality in the Δ⅓-degrees.Philip Welch - 1987 - Journal of Symbolic Logic 52 (4):908-915.
  33. Set-Theoretic Absoluteness and the Revision Theory.Philip Welch - 2003 - Bulletin of Symbolic Logic 9 (2):235-237.
  34.  8
    Stably measurable cardinals.Philip D. Welch - 2021 - Journal of Symbolic Logic 86 (2):448-470.
    We define a weak iterability notion that is sufficient for a number of arguments concerning $\Sigma _{1}$ -definability at uncountable regular cardinals. In particular we give its exact consistency strength first in terms of the second uniform indiscernible for bounded subsets of $\kappa $ : $u_2$, and secondly to give the consistency strength of a property of Lücke’s.TheoremThe following are equiconsistent:There exists $\kappa $ which is stably measurable;for some cardinal $\kappa $, $u_2=\sigma $ ;The $\boldsymbol {\Sigma }_{1}$ -club property holds (...)
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  35.  33
    The natural hierarchy and quasi-hierarchy of constructibility degrees.Philip Welch - 1986 - Journal of Symbolic Logic 51 (1):130-134.
    We investigate the set S 2 of "quickly sharped" reals: \begin{align*}S_2 &= \{x \mid x \in M, M \text{the} <^\ast-\text{least mouse} \not\in L\lbrack x\rbrack\} \\ &= \{x \mid L\lbrack x\rbrack \models "V = K"\},\\ \end{align*} in the manner of [K] defining a natural hierarchy and quasi-hierarchy of constructibility degrees and identifying their termination points.
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  36.  12
    The ramified analytical hierarchy using extended logics.Philip D. Welch - 2018 - Bulletin of Symbolic Logic 24 (3):306-318.
    The use of Extended Logics to replace ordinary second order definability in Kleene’s Ramified Analytical Hierarchy is investigated. This mirrors a similar investigation of Kennedy, Magidor and Väänänen [11] where Gödel’s universe L of constructible sets is subjected to similar variance. Enhancing second order definability allows models to be defined which may or may not coincide with the original Kleene hierarchy in domain. Extending the logic with game quantifiers, and assuming strong axioms of infinity, we obtain minimal correct models of (...)
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  37.  27
    The reals in core models.Philip Welch - 1987 - Journal of Symbolic Logic 52 (1):64-67.
    We set $\mathscr{D} = \langle\mathscr{D}, \leq_L, \tt\#\rangle$ , where D is the set of degrees of nonconstructibility for countable sets of countable ordinals. We show how to define inductively over this structure the degrees of such sets of ordinals in K, the core model, and the next few core models thereafter, i.e. without reference to mice, premice or measurable cardinals.
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  38.  18
    Menachem Magidor. Representing sets of ordinals as countable unions of sets in the core model. Transactions of the American Mathematical Society, vol. 317 , pp. 91–126. [REVIEW]Philip Welch - 1995 - Journal of Symbolic Logic 60 (2):701-704.
  39.  6
    Review: Menachem Magidor, Representing Sets of Ordinals as Countable Unions of Sets in the Core Model. [REVIEW]Philip Welch - 1995 - Journal of Symbolic Logic 60 (2):701-704.
  40.  14
    Review: Sy D. Friedman, Strong Coding; Sy D. Friedman, A Guide to "Strong Coding.". [REVIEW]Philip Welch - 1990 - Journal of Symbolic Logic 55 (3):1311-1313.
  41.  4
    Sy D. Friedman. Strong coding. Annals of pure and applied logic, vol. 35, pp. 1–98. - Sy D. Friedman. A guide to “Strong coding.”Annals of pure and applied logic, vol. 35, pp. 99–122. [REVIEW]Philip Welch - 1990 - Journal of Symbolic Logic 55 (3):1311-1313.
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