In February of 2007, I converted my “What’s new” web page of research
updates into a blog at 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 .
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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