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

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