I read this for the first time in the excellent and surprisingly accessible The Annotated Turing[0], which I can highly recommend. If you're vaguely interested in things like proofs like these or about computability or just Turing's and others' contributions and approaches, the book approaches these things very well without presuming a deep pure mathematical background.

Seriously, read it.

[0] http://www.amazon.com/Annotated-Turing-Through-Historic-Comp...

Another good book by Charles Petzold is The Annotated Turing.

http://www.amazon.com/Annotated-Turing-Through-Historic-Comp...