Book123

    Free register and download UseNet downloader, then you can FREE Download from UseNet.

    Download without Limit " Science/Engineering Proof Theory " from UseNet for FREE!

More

Gaisi Takeuti, "Proof Theory"

From reader's review:

Takeuti was undoubtedly one of the greatest logicians of the twentieth century and this book is indispensable reading for anyone who wishes fully to appreciate his contributions. The first three chapters are also quite a good introduction to proof theory in general. On the whole, although the second and third chapters do contain a fair number of theorems first proved by Takeuti himself, the book can pretty much be divided into two halves: the first three chapters present standard results of proof theory while the fourth, fifth and sixth are devoted almost entirely to Takeuti’s own work.

The first chapter introduces Gentzen’s sequent calculi LJ and LK and proves Gentzen’s cut-elimination theorem as well as completeness theorems for both LJ and LK.

The centrepiece of the second chapter is Gentzen’s second (1938) proof of the eliminability of essential cuts from a derivation of the empty sequent in PA. Takeuti’s presentation of the proof is very helpful and makes some things clearer than Gentzen’s own presentation. This proof also constitutes the starting point for Takeuti’s own cut-elimination proofs. On the other hand, Takeuti omits the helpful heuristic discussion which accompanies Gentzen’s presentation of the proof, so someone wishing to get to know the proof really well would do best to read both Gentzen’s and Takeuti’s versions. In any case, a thorough knowledge of this proof is essential before you go on to the higher-level stuff.

The third chapter is good as an introduction to general second-order logic and simple type theory, culminating in the proof of cut-eliminability via completeness for simple type theory, which was produced by Takahashi and Prawitz simultaneously, in the late 1960’s.

I skipped the very long fourth chapter, so cannot really comment on its contents. The purpose of the chapter is, by considering infinitary logic, to enable familiar theorems of model theory to be re-stated as theorems of proof theory.

With the fifth chapter, we come to the material which Takeuti is mainly famous for: the cut-elimination proof for the Pi-1-1 sub-systems of second order logic and arithmetic. It is notoriously difficult to see the strategic thinking behind this proof and for this reason Takeuti’s methods never caught on much outside Japan.

The difficulty of perceiving the ideas behind these proofs is not helped by the absence from the book of almost any heuristic discussion. I would, in fact, not advise anyone wanting to understand this proof to begin with the present book. There is a sequence of articles by Takeuti called “On the Fundamental Conjecture of GLC” numbered I through to VI, which appeared mainly in the Journal of the Mathematical Society of Japan in the 1950’s and it is very helpful to read these articles in chronological order as, in doing so, you see Takeuti gradually extending his techniques so as to deal with stronger and stronger calculi...

Buy Book at Lowest Price on Amazon

Not all books ... appear on the homepage.

In order not to miss many of them follow Download Link (Ebooks section) (see top of each page on AH)

and visit too :)

Rating:
2.5 out of 5 by Book123