Aspects of Diagonalization & Provability
Albert Visser
Abstract:
Aspects of Diagonalization & Provability
Albert Visser
Monolithic this thesis is not. It reflects the history of our preoccupation with Gödel’s Incompleteness Theorems over the last four years.
In part 1 and 2 we concentrate on generalizations of Gödel’s First Incompleteness Theorem like the Gödel-Rosser-MostowskiMyhill-Kripke Theorem [GRMMK for short]. One cannot but be impressed how few specific facts about formal theories are needed to prove this theorem (just consistency and a weak kind of representability of the partial recursive functions). The rest is done by recursion theory; one is tempted to say that here recursion theory plays the role of "general nonsense" with respect to the theory of formal systems. An obvious question in this context is whether the applications of recursion theory to formal systems can be made uniform, whether one can give recursion theoretic results in "the right form" for these applications.
In part 1 and 2 tentative answers are considered. In part 1 we adapt Ershov’s theory of Numerations for our purpose. A disadvantage of the framework thus obtained is this: when one analyses an application to a formal system it sometimes turns out that there is a direct "theory-free" proof in which less about the system is used than demanded by the framework. For example in certain applications to arithmetic one uses the existence of the \Sigma^0_n-truthpredicate, to lambda-calculus the existence of the universal constructor E. These facts are not needed for your average recursive inseparability argument. So perhaps still some improvement is possible.. All the same the framework is rather natural for a wide range of applications.
As so many, we were at first victims of that easy, youthful fallacy: to think that Gödel's Second Incompleteness Theorem is "just" a formalization of Gödel’s First Incompleteness Theorem and that therefore Gödel’s Second Incompleteness Theorem does not differ programmatically from Gödel's First Incompleteness Theorem. (Clearly this fallacy is just an instance of a far more general fallacy, and is ultimately connected with a wrong appreciation of the very aims of formalization.) In due time we saw the fallacy’s true nature, mainly because of Craig Smorynski’s benificial influence. The strength of CRMMK is also a weakness, when looked upon as a basis for further work. Because it is so much a recursion theoretic result one is encouraged to forget the intrinsic interest of the concepts of formal provability and concistency.
Consequently part 3,4 and 5 are devoted to the concept of formal provability, or to be more specific to Provability Logic. Provability Logic is of course not the only possible approach to study formal provability (there is e.g. the work of Jeroslow], but we think its results, in particular De Jongh’s Theorem and Solovay’s Completeness Theorem are at present the deepest ones pertaining specifically to formal provability. Solovay’s Theorem seems the natural generalization of Gödel’s Second Incompleteness Theorem at least from "a propositional point of view".
Part 4 adds a new element: it is on the provability logic of intuitionistic arithmetic. Maybe this is the proper place to state our view on Intuitionism / Constructivism. We think classical mathematics is just plainly true, e.g. Format’s Theorem is either true or false. Even if it is very well possible that it is true but not provable in principle for human beings. In other words that it is simply outside the scope of the humana priori. (Can the mathematical a priori for Martians be stronger at some points than ours?) Still Constructivism is a perFectly legitimate branch of Mathematics like Numerical Analysis, Homological Algebra or Algebraic Geometry. It could be considered as that branch of mathematics that does systematically ("globally") what in other branches is done case for case ("locally"): finding algorithms corresponding to Existence Theorems (this ”hackneyed business" of numerical content). A further point is that mathematicians are often interested in theories with many models. Now theories like Classical Arithmetic and Analysis suffer from a lack of (reasonable) models. In the Intuitionistic/ Constructivistic case however there are many models. In current developments in Intuitionistic Model Theory people try to put this aspect to good use. At present it is impossible to judge the merits of topos theory as intuitionistic model theory, but surely there is more than a grain of a good idea behind that approach.
The remaining section of the Introductory Part is an exposition of Solovay’s Completeness Theorem, to introduce the reader to parts 3, 4, 5. The thesis is concluded by an Epilogue on the provability logic of Heyting’s Arithmetic, both to elaborate on the results of part 4 and to provide some context for them.