Statement of Research Interests
نویسنده
چکیده
Generic absoluteness. Paul Cohen’s work in the 1960s struck a blow against the idea of mathematics as absolute truth. Cohen showed that there is a natural mathematical statement, namely the Continuum Hypothesis, whose truth cannot be proved or disproved in ZFC. (Gödel sentences were discovered earlier but are arguably not natural mathematical statements.) Cohen’s method, called forcing, enlarges a given model of set theory by adjoining an ideal “generic” object, producing a new model that typically exhibits a different set theory. Applications of this method now constitute a major area of research. However, there are limits to what it can do. For example, forcing cannot change the truth of arithmetic statements simply because it cannot change the structure (N; 0, 1,+,×) of arithmetic. Accordingly, we say that arithmetic statements are “generically absolute.” As we go up the complexity hierarchy, the nature of mathematical truth undergoes a transition from absolute (such as arithmetic) to relative (such as the Continuum Hypothesis.) For statements about real numbers, the question of generic absoluteness is quite subtle and is tied to the existence of tree representations for sets of real numbers. A tree representation for a set A of “logician’s reals” (integer sequences) is a continuous function assigning to every real x a tree Tx such that x ∈ A if and only if Tx has an infinite branch. The existence of infinite branches is absolute, and the classical absoluteness theorems of Mostowski and Shoenfield can be proved using the existence of tree representations for Σ1 and Σ 1 2 sets of reals respectively. For more complex statements about real numbers, the connection between generic absoluteness phenomena and tree representations persists for a while, but now to get a complete picture we need to go beyond ZFC and introduce large cardinals. This is because while the trees for Σ1 sets of reals are countable and the trees for Σ2 sets of reals have size א1, the trees for more complicated sets typically have very large cardinality. For example, Σ3 sets of reals are represented by trees of size equal to a measurable cardinal if one
منابع مشابه
Correction: Implications of Hybridization, NUMTs, and Overlooked Diversity for DNA Barcoding of Eurasian Ground Squirrels
The Funding statement was erroneously left out of this publication. The publisher apologizes for this error. The Funding statement should read: "This research was supported by the Russian Foundation for Basic Research (RFBR; grant 14-04-00301) to SVT. Funding for DNA barcoding analysis at the Biodiversity Institute of Ontario was provided by grants from the Natural Sciences and Engineering Rese...
متن کاملJean Gourd Statement of Research
I believe that research and teaching collectively support one another. Research strengthens teaching; teaching sustains research. As a result, a lot of what drives my research is not only based on my personal interests within the field of computer science, but is also influenced by what I teach and, perhaps more specifically, the feedback that I receive from the students that I teach. I find th...
متن کاملPersonal Statement and Outline of Proposed Research to support a PhD application
My primary research interests lie in the fields of operating systems, distributed systems and programming languages. I also have interests in software engineering, networks, continuous media applications, sentient environments and human-computer interaction. I intend to pursue a career in research, and during 2005–6 have been a Research Assistant at the Computer Laboratory of the University of ...
متن کاملStatement on Research
My research interests are primarily in software testing, with a focus on automated strategies and empirical methodologies for web application testing. I am also broadly interested in software engineering and distributed systems. In this statement, I present an overview of my doctoral thesis, extensions to my thesis, future research directions, and my experiences and philosophy on mentoring unde...
متن کاملStatement of Research Interests and Goals
My research interests and activities over the past six years have mainly centered around three areas: Databases, Middleware for Distributed and Mobile Applications, and Hybrid Intelligent Systems and its Applications. In addition to these research areas, I was also actively involved in a NSF funded "Digital Library" project that created a central repository of educational materials for Computer...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2014