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  1. P. D. Welch, Global Reflection Principles.
    Reflection Principles are commonly thought to produce only strong axioms of infinity consistent with V = L. It would be desirable to have some notion of strong reflection to remedy this, and we have proposed Global Reflection Principles based on a somewhat Cantorian view of the universe. Such principles justify the kind of cardinals needed for, inter alia , Woodin’s Ω-Logic.
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  2. P. D. Welch (forthcoming). Some Observations on Truth Hierarchies. Review of Symbolic Logic:1-30.
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  3. Sy-David Friedman & P. D. Welch (2011). Hypermachines. Journal of Symbolic Logic 76 (2):620 - 636.
    The Infinite Time Turing Machine model [8] of Hamkins and Kidder is, in an essential sense, a "Σ₂-machine" in that it uses a Σ₂ Liminf Rule to determine cell values at limit stages of time. We give a generalisation of these machines with an appropriate Σ n rule. Such machines either halt or enter an infinite loop by stage ζ(n) = df μζ(n)[∃Σ(n) > ζ(n) L ζ(n) ≺ Σn L Σ(n) ], again generalising precisely the ITTM case. The collection of (...)
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  4. I. Sharpe & P. D. Welch (2011). Greatly Erdős Cardinals with Some Generalizations to the Chang and Ramsey Properties. Annals of Pure and Applied Logic 162 (11):863-902.
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  5. P. D. Welch (2011). Determinacy in Strong Cardinal Models. Journal of Symbolic Logic 76 (2):719 - 728.
    We give limits defined in terms of abstract pointclasses of the amount of determinacy available in certain canonical inner models involving strong cardinals. We show for example: Theorem A. $\mathrm{D}\mathrm{e}\mathrm{t}\text{\hspace{0.17em}}({\mathrm{\Pi }}_{1}^{1}-\mathrm{I}\mathrm{N}\mathrm{D})$ ⇒ there exists an inner model with a strong cardinal. Theorem B. Det(AQI) ⇒ there exist type-1 mice and hence inner models with proper classes of strong cardinals. where ${\mathrm{\Pi }}_{1}^{1}-\mathrm{I}\mathrm{N}\mathrm{D}\phantom{\rule{0ex}{0ex}}$ (AQI) is the pointclass of boldface ${\mathrm{\Pi }}_{1}^{1}$ -inductive (respectively arithmetically quasi-inductive) sets of reals.
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  6. P. D. Welch (2011). Truth, Logical Validity and Determinateness: A Commentary on Field's Saving Truth From Paradox. Review of Symbolic Logic 4 (3):348-359.
    We consider notions of truth and logical validity defined in various recent constructions of Hartry Field. We try to explicate his notion of determinate truth by clarifying the path-dependent hierarchies of his determinateness operator.
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  7. P. D. Welch (2011). Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions. Journal of Symbolic Logic 76 (2):418 - 436.
    We locate winning strategies for various ${\mathrm{\Sigma }}_{3}^{0}$ -games in the L-hierarchy in order to prove the following: Theorem 1. KP+Σ₂-Comprehension $\vdash \exists \alpha L_{\alpha}\ models"\Sigma _{2}-{\bf KP}+\Sigma _{3}^{0}-\text{Determinacy}."$ Alternatively: ${\mathrm{\Pi }}_{3}^{1}\text{\hspace{0.17em}}-{\mathrm{C}\mathrm{A}}_{0}\phantom{\rule{0ex}{0ex}}$ "there is a β-model of ${\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17 em}}{\mathrm{\Sigma }}_{3}^{0}$ -Determinacy." The implication is not reversible. (The antecedent here may be replaced with ${\mathrm{\Pi }}_{3}^{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({\mathrm{\Pi }}_{3}^{1}\right)-{\mathrm{C}\mathrm{A}}_{0}:\text{\hspace{0.17em}}{\mathrm{\Pi }}_{3}^{1}$ instances of Comprehension with only ${\mathrm{\Pi }}_{3}^{1}$ -lightface definable parameters—or even weaker theories.) Theorem 2. KP +Δ₂-Comprehension +Σ₂-Replacement + ${\mathrm{\Sigma }}_{3}^{0}\phantom{\rule{0ex}{0ex}}$ -Determinacy. (Here AQI (...)
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  8. P. D. Welch (2009). Games for Truth. Bulletin of Symbolic Logic 15 (4):410-427.
    We represent truth sets for a variety of the well known semantic theories of truth as those sets consisting of all sentences for which a player has a winning strategy in an infinite two person game. The classifications of the games considered here are simple, those over the natural model of arithmetic being all within the arithmetical class of $\Sum_{3}^{0}$.
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  9. P. D. Welch (2008). The Extent of Computation in Malament–Hogarth Spacetimes. British Journal for the Philosophy of Science 59 (4):659-674.
    We analyse the extent of possible computations following Hogarth ([2004]) conducted in Malament–Hogarth (MH) spacetimes, and Etesi and Németi ([2002]) in the special subclass containing rotating Kerr black holes. Hogarth ([1994]) had shown that any arithmetic statement could be resolved in a suitable MH spacetime. Etesi and Németi ([2002]) had shown that some relations on natural numbers that are neither universal nor co-universal, can be decided in Kerr spacetimes, and had asked specifically as to the extent of computational limits there. (...)
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  10. P. D. Welch (2008). Ultimate Truth Vis- à- Vis Stable Truth. Review of Symbolic Logic 1 (1):126-142.
    We show that the set of ultimately true sentences in Hartry Field's Revenge-immune solution model to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger's revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second-order number theory is needed to establish the semantic values of sentences in Field's relative consistency proof of his theory over the ground model of the standard natural numbers: -CA0 (...)
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  11. Agustín Rayo & P. D. Welch (2007). Field on Revenge. In J. C. Beall (ed.), Revenge of the Liar: New Essays on the Paradox. Oxford University Press.
     
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  12. P. D. Welch (2005). Some Open Problems in Mutual Stationarity Involving Inner Model Theory: A Commentary. Notre Dame Journal of Formal Logic 46 (3):375-379.
    We discuss some of the relationships between the notion of "mutual stationarity" of Foreman and Magidor and measurability in inner models. The general thrust of these is that very general mutual stationarity properties on small cardinals, such as the ℵns, is a large cardinal property. A number of open problems, theorems, and conjectures are stated.
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  13. P. D. Welch (2004). On Unfoldable Cardinals, Ω-Closed Cardinals, and the Beginning of the Inner Model Hierarchy. Archive for Mathematical Logic 43 (4):443-458.
    Let κ be a cardinal, and let H κ be the class of sets of hereditary cardinality less than κ ; let τ (κ) > κ be the height of the smallest transitive admissible set containing every element of {κ}∪H κ . We show that a ZFC-definable notion of long unfoldability, a generalisation of weak compactness, implies in the core model K, that the mouse order restricted to H κ is as long as τ. (It is known that some weak (...)
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  14. J. D. Hamkins & P. D. Welch (2003). Pf= NPf Almost Everywhere. Mathematical Logic Quarterly 49 (5):536-540.
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  15. P. D. Welch (2003). On Revision Operators. Journal of Symbolic Logic 68 (2):689-711.
    We look at various notions of a class of definability operations that generalise inductive operations, and are characterised as “revision operations”. More particularly we: (i) characterise the revision theoretically definable subsets of a countable acceptable structure; (ii) show that the categorical truth set of Belnap and Gupta’s theory of truth over arithmetic using \emph{fully varied revision} sequences yields a complete \Pi13 set of integers; (iii) the set of \emph{stably categorical} sentences using their revision operator ψ is similarly \Pi13 and which (...)
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  16. J. Vickers & P. D. Welch (2001). On Elementary Embeddings From an Inner Model to the Universe. Journal of Symbolic Logic 66 (3):1090-1116.
    We consider the following question of Kunen: Does Con(ZFC + ∃M a transitive inner model and a non-trivial elementary embedding j: M $\longrightarrow$ V) imply Con (ZFC + ∃ a measurable cardinal)? We use core model theory to investigate consequences of the existence of such a j: M → V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of "there exists a proper class of almost Ramsey cardinals". Conversely, (...)
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  17. P. D. Welch (2001). On Gupta-Belnap Revision Theories of Truth, Kripkean Fixed Points, and the Next Stable Set. Bulletin of Symbolic Logic 7 (3):345-360.
    We consider various concepts associated with the revision theory of truth of Gupta and Belnap. We categorize the notions definable using their theory of circular definitions as those notions universally definable over the next stable set. We give a simplified (in terms of definitional complexity) account of varied revision sequences-as a generalised algorithmic theory of truth. This enables something of a unification with the Kripkean theory of truth using supervaluation schemes.
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  18. J. Vickers & P. D. Welch (2000). On Successors of Jónsson Cardinals. Archive for Mathematical Logic 39 (6):465-473.
    We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero [4] calculates correctly the successors of Jónsson cardinals, assuming $O^{Sword}$ does not exist. Namely, if $\kappa$ is a Jónsson cardinal then $\kappa^+ = \kappa^{+K}$ , provided that there is no non-trivial elementary embedding $j:K \longrightarrow K$ . There are a number of related results in ZFC concerning $\cal{P}(\kappa)$ in V and inner models, for $\kappa$ a Jónsson or singular cardinal.
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  19. P. D. Welch (2000). Eventually Infinite Time Turing Machine Degrees: Infinite Time Decidable Reals. Journal of Symbolic Logic 65 (3):1193-1203.
    We characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated (...)
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  20. P. D. Welch (1997). Review: William J. Mitchell, John R. Steel, Fine Structure and Iteration Trees. [REVIEW] Journal of Symbolic Logic 62 (4):1491-1493.
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  21. P. D. Welch (1996). Countable Unions of Simple Sets in the Core Model. Journal of Symbolic Logic 61 (1):293-312.
    We follow [8] in asking when a set of ordinals $X \subseteq \alpha$ is a countable union of sets in K, the core model. We show that, analogously to L, and X closed under the canonical Σ 1 Skolem function for K α can be so decomposed provided K is such that no ω-closed filters are put on its measure sequence, but not otherwise. This proviso holds if there is no inner model of a weak Erdős-type property.
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  22. P. D. Welch (1996). Determinacy in the Difference Hierarchy of Co-Analytic Sets. Annals of Pure and Applied Logic 80 (1):69-108.
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  23. P. D. Welch (1994). Characterising Subsets of Ω1 Constructible From a Real. Journal of Symbolic Logic 59 (4):1420 - 1432.
    A small large cardinal upper bound in V for proving when certain subsets of ω 1 (including the universally Baire subsets) are precisely those constructible from a real is given. In the core model we find an exact equivalence in terms of the length of the mouse order; we show that $\forall B \subseteq \omega_1 \lbrack B$ is universally Baire $\Leftrightarrow B \in L\lbrack r \rbrack$ for some real r] is preserved under set-sized forcing extensions if and only if there (...)
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  24. P. D. Welch (1988). Some Descriptive Set Theory and Core Models. Annals of Pure and Applied Logic 39 (3):273-290.
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