Preface In February of 2007, I converted my “What’s new” web page of research updates into a blog at terrytao.wordpress.com. This blog has since grown and evolved to cover a wide variety of mathematical topics, ranging from my own research updates, to lectures and guest posts by other mathematicians, to open problems, to class lecture notes, to expository articles at both basic and advanced levels. In 2010, I also started writing shorter mathematical articles on my Google Buzz feed at profiles.google.com/114134834346472219368/buzz . This book collects some selected articles from both my blog and my Buzz feed from 2010, continuing a series of previous books [Ta2008], [Ta2009], [Ta2009b], [Ta2010], [Ta2010b], [Ta2011], [Ta2011b], [Ta2011c] based on the blog. The articles here are only loosely connected to each other, although many of them share common themes (such as the titular use of compactness and contradiction to connect finitary and infinitary mathematics to each other). I have grouped them loosely by the general area of mathematics they pertain to, although the dividing lines between these areas is somewhat blurry, and some articles arguably span more than one category. Each chapter is roughly organised in increasing order of length and complexity (in particular, the first half of each chapter is mostly devoted to the shorter articles from my Buzz feed, with the second half comprising the longer articles from my blog). A remark on notation For reasons of space, we will not be able to define every single mathematical term that we use in this book. If a term is italicised for reasons other than xi

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