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4A Gitik Iteration With Nearly Easton FactoringJournal of Symbolic Logic 68 (2): 481-502. 2003.We reprove Gitik’s theorem that if the GCH holds and $o=\gk+1$ then there is a generic extension in which $\gk$ is still measurable and there is a closed unbounded subset C of $\gk$ such that every $ν\in C$ is inaccessible in the ground model. Unlike the forcing used by Gitik, the iterated forcing $\radin\gl+1$ used in this paper has the property that if $\gl$ is a cardinal less then $\gk$ then $\radin\gl+1$ can be factored in V as $\radin\gk+1=\radin\gl+1\times\radin\gl+1,\gk$ where $\card{\rad…Read more
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36Medium Theory: Preface to the 2003 "Critical Inquiry" SymposiumCritical Inquiry 30 (2): 324. 2004.
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27D. A. Martin and J. R. Steel. Iteration trees. Journal of the American Mathematical Society, vol. 7 , pp. 1–73 (review)Bulletin of Symbolic Logic 8 (4): 545-546. 2002.
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35Does God roll dice? Neutrality and determinism in evolutionary ecologyBiology and Philosophy 34 (1): 3. 2019.A tension between perspectives that emphasize deterministic versus stochastic processes has sparked controversy in ecology since pre-Darwinian times. The most recent manifestation of the contrasting perspectives arose with Hubbell’s proposed “neutral theory”, which hypothesizes a paramount role for stochasticity in ecological community composition. Here we shall refer to the deterministic and the stochastic perspectives as the niche-based and neutral-based research programs, respectively. Our go…Read more
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19The sharp for the Chang model is smallArchive for Mathematical Logic 56 (7-8): 935-982. 2017.Woodin has shown that if there is a measurable Woodin cardinal then there is, in an appropriate sense, a sharp for the Chang model. We produce, in a weaker sense, a sharp for the Chang model using only the existence of a cardinal \ having an extender of length \.
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I[ω₂] can be the nonstationary ideal on Cof. Transactions of the American Mathematical Society, vol. 361Bulletin of Symbolic Logic 17 (4): 535-537. 2011.
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41On the Hamkins approximation propertyAnnals of Pure and Applied Logic 144 (1-3): 126-129. 2006.We give a short proof of a lemma which generalizes both the main lemma from the original construction in the author’s thesis of a model with no ω2-Aronszajn trees, and also the “Key Lemma” in Hamkins’ gap forcing theorems. The new lemma directly yields Hamkins’ newer lemma stating that certain forcing notions have the approximation property
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16Review: D. A. Martin, J. R. Steel, Iteration Trees (review)Bulletin of Symbolic Logic 8 (4): 545-546. 2002.
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49Adding Closed Unbounded Subsets of ω₂ with Finite ForcingNotre Dame Journal of Formal Logic 46 (3): 357-371. 2005.An outline is given of the proof that the consistency of a κ⁺-Mahlo cardinal implies that of the statement that I[ω₂] does not include any stationary subsets of Cof(ω₁). An additional discussion of the techniques of this proof includes their use to obtain a model with no ω₂-Aronszajn tree and to add an ω₂-Souslin tree with finite conditions
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192A weak variation of Shelah's I[ω₂]Journal of Symbolic Logic 69 (1): 94-100. 2004.We use a $\kappa^{+}-Mahlo$ cardinal to give a forcing construction of a model in which there is no sequence $\langle A_{\beta} : \beta \textless \omega_{2} \rangle$ of sets of cardinality $\omega_{1}$ such that $\{\lambda \textless \omega_{2} : \existsc \subset \lambda & (\bigcupc = \lambda otp(c) = \omega_{1} & \forall \beta \textless \lambda (c \cap \beta \in A_{\beta}))\}$ is stationary
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198A Gitik iteration with nearly Easton factoringJournal of Symbolic Logic 68 (2): 481-502. 2003.We reprove Gitik's theorem that if the GCH holds and o(κ) = κ + 1 then there is a generic extension in which κ is still measurable and there is a closed unbounded subset C of κ such that every $\nu \in C$ is inaccessible in the ground model. Unlike the forcing used by Gitik. the iterated forcing $R_{\lambda +1}$ used in this paper has the property that if λ is a cardinal less then κ then $R_{\lambda + 1}$ can be factored in V as $R_{\kappa + 1} = R_{\lambda + 1} \times R_{\lambda + 1, \kappa}$ w…Read more
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University of Alabama, BirminghamUndergraduate
Birmingham, Alabama, United States of America