Statement Chris Lambie
نویسنده
چکیده
My research lies mostly in logic and set theory, and in applications of set-theoretic tools to other areas of mathematics, such as graph theory, algebra, and topology. My set-theoretic work is largely combinatorial in nature and comes in one of two flavors: ZFC results and independence results. ZFC stands for Zermelo-Fraenkel axioms with choice and is the standard set of axioms in which set theory is done. Many interesting set-theoretic statements can be proven outright from the axioms of ZFC. However, it has been known since the 1930s [12] that, given any sufficiently strong, effectively axiomatizable formal system (such as ZFC), there are statements that can be neither proven nor disproven within the system. In 1963, Cohen [4] introduced the method of forcing, which, together with earlier work of Gödel [13], allowed him to show that the Continuum Hypothesis, the most famous open problem of set theory at the time, is undecidable by the axioms of ZFC. Since then, forcing, which allows one to construct new models of ZFC by introducing certain “generic” sets, has become a central technique in set theory, and I make use of it extensively in my work. Another central concept in modern set theory, and in my work on independence results, is the notion of a large cardinal. Roughly speaking, a large cardinal is a type of infinite cardinal number whose consistent existence is not implied by ZFC. For example, a weakly compact cardinal is an uncountable cardinal κ for which the following generalization of Ramsey’s theorem holds:
منابع مشابه
Bounded Stationary Reflection
We prove that, assuming large cardinals, it is consistent that there are many singular cardinals μ such that every stationary subset of μ+ reflects but there are stationary subsets of μ+ that do not reflect at ordinals of arbitrarily high cofinality. This answers a question raised by Todd Eisworth.
متن کاملOn the strengths and weaknesses of weak squares
• Let T be a countable theory with a distinguished predicate R. A model of T is said to be of type (λ,μ) if the cardinality of the model is λ and the cardinality of the model’s interpretation of R is μ. For cardinals α,β, γ, and δ, (α,β) → (γ, δ) is the assertion that for every countable theory T , if T has a model of type (α,β), then it has a model of type (γ, δ). Chang showed that under GCH, ...
متن کاملDiagonal Supercompact Radin Forcing
In this paper we consider two methods for producing models with some global behavior of the continuum function on singular cardinals and the failure of weak square. The first method is as an extension of Sinapova’s work [21]. We define a diagonal supercompact Radin forcing which adds a club subset to a cardinal κ while forcing the failure of SCH everywhere on the club. The intuition from Sinapo...
متن کاملThe Hanf number for Amalgamation of Coloring Classes
We study amalgamation properties in a family of abstract elementary classes that we call coloring classes. The family includes the examples previously studied in [3]. We establish that the amalgamation property is equivalent to the disjoint amalgamation property in all coloring classes; find the Hanf number for the amalgamation property for coloring classes; and improve the results of [3] by sh...
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تاریخ انتشار 2017