Craig Smorynski

Self-Reference and Modal Logic

0. Introduction.- 1. The Incompleteness Theorems.- 2. Self-Reference.- 3. Things to Come.- 4. The Theory PRA.- 5. Encoding Syntax in PRA.- 6. Additional Arithmetic Prerequisites.- I. The Logic of Provability.- 1. Provability as Modality.- 1. A System of Basic Modal Logic.- 2. Provability Logic(s).- 3. Self-Reference in PRL.- 4. Avoiding R2.- 2. Modal Model Theory.- 1. Model Theory for BML.- 2. Model Theory for PRL.- 3. Models and Self-Reference.- 4. Another Provability Logic.- 3. Arithmetic Interpretations of PRL.- 1. Solovays First Completeness Theorem.- 2. Solovays Second Completeness Theorem.- 3. Generalisations, Refinements, and Analogues.- II. Multi-Modal Logic and Self-Reference.- 4. Bi-Modal Logics and Their Arithmetic Interpretations.- 1. Bi-Modal Self-Reference.- 2. Kripke Models.- 3. Carlson Models.- 4. Carlsons Arithmetic Completeness Theorem.- 5. Fixed Point Algebras.- 1. Boolean and Diagonalisable Algebras.- 2. Fixed Point Algebras.- 3. Discussion.- III. Non-Extensional Self-Reference.- 6. Rosser Sentences.- 1. Modal Systems for Rosser Sentences.- 2. Arithmetic Interpretations.- 3. Inequivalent Rosser Sentences.- 7. An Ubiquitous Fixed Point Calculation.- 1. An Ubiquitous Fixed Point Calculation.- 2. Applications.- 3. Relativisation to a Partial Truth Definition.- 4. Svejdars Self-Referential Formulae.

Fifty years of self-reference in arithmetic.

It is now fifty years since Hans Hahn first presented an abstract of the then unknown Kurt Godel to the Vienna Academy of Sciences. The rest, as it is said, is history. Much of this history is well-known and I do not propose to repeat the usual platitudes. On any golden anniversary, however, it is natural to look back and I am not one to rebel against nature. On this occasion I will sing the hitherto unsung song of diagonalisation. While self-reference is one of the more outstanding features of GodePs work and self-reference in arithmetic has had some no-table success, this success is neither so widely known nor so great that my message should bore the average reader. One of the great curiosities of my topic is how long it took (perhaps better: is taking) for the subject to develop. Even the most obvious and central fact—the Diagonalisation Theorem—seems to have had difficulty in surfacing. It is not to be found in many of the basic textbooks (e.g., Kleene [20], Mendelson [25], Shoenfield [38], Bell and Machover [1], and Manin [24]) and it is only stated in its most rudimentary form in most others (e.g., Boolos and Jeffrey [31, Enderton [4], and Monk [26]). The two most substantial expositions of Incompleteness Theory (Mostowski [28] and Stegmuller [47]) offer no explicit statement of the Diagonalisation Theorem in any form. Indeed, it is only in a recent more advanced exposition (Boolos [2]) that the full Diagonalisation Theorem has finally graced the pages of a book. Yet, diagonalisation in arithmetic is fifty years old and was stated in proper generality in print in 1962 [27]—long before most of the available textbooks were written. Perhaps, before writing another word, I should outline the history of the Diagonalisation Theorem. While I have not made an exhaustive search of the literature, I can report that a cursory examination of the more important papers yields the following development:

Beth's Theorem and Self-Referential Sentences

Publisher Summary This chapter discusses Beths theorem and self-referential sentences. This chapter presents a new proof of Beths theorem. This theorem is most conveniently stated in the notation of modal logic and is best viewed as a theorem about modal logic. The Interpolation Theorem for L is proved and Beths Theorem is obtained from it. The Interpolation Theorem is model-theoretic and is based on R. Solovays proof of the completeness theorem for L with respect to Kripke models. This is followed by de Jonghs proof of the existence and uniqueness result. The classes of self-referential sentences considered in the chapter are those that correspond to fixed-points of appropriate modal functions. The proof in the chapter is a pure existence proof and does not result in an actual explicit definition of the fixed-point.

Calculating self-referential statements, I: Explicit calculations

The proof of the Second Incompleteness Theorem consists essentially of proving the uniqueness and explicit definability of the sentence asserting its own unprovability. This turns out to be a rather general phenomenon: Every instance of self-reference describable in the modal logic of the standard proof predicate obeys a similar uniqueness and explicit definability law. The efficient determination of the explicit definitions of formulae satisfying a given instance of self-reference reduces to a simple algebraic problem-that of solving the corresponding fixed-point equation in the modal logic. We survey techniques for the efficient calculation of such fixed-points.

The finite inseparability of the first-order theory of diagonalisable algebras

In a recent paper, Montagna proved the undecidability of the first-order theory of diagonalisable algebras. This result is here refined — the set of finitely refutable sentences is shown effectively inseparable from the set of theorems. The proof is quite simple.

A note on the number of zeros of polynomials and exponential polynomials

?0. The negative solution of Hilberts Tenth Problem brought with it a number of unsolvable Diophantine problems. Moreover, by actually providing a Diophantine characterization of recursive enumerability, the proof of the negative solution opened the door to the techniques of recursion theory. In this note, we wish to apply several recursion-theoretic facts and an improvement on the exponential Diophantine representation to refine the exponential case of a result of Davis [1972] regarding the difficulty of determining the number of zeros of a polynomial. P, Q, etc. will denote polynomials or exponential polynomials-exactly which will be clear from the context. Let # (P) denote the number of distinct nonnegative zeros of P. Further, let C = {0, 1, * * *, N4o} be the set of possible values of # (P). For A C C, we define A * to be

Calculating Self-Referential Statements: Guaspari Sentences of the First Kind

Beginning in 1960, and continuing for about a decade and a half, Shepherdsons self-referential formulae dominated the applications of diagonalization in metamathematics. In 1976, it was doubly toppled from its position of supremacy by two demonstrably more powerful such sentences introduced by D. Guaspari. The goal of the present note is to understand and explain the first (and, for the time being, more important) of Guasparis self-referential sentences, which we dub “Guaspari sentences of the first kind”, or, less poetically, “Guaspari fixed points”. Both Shepherdsons and Guasparis fixed points are generalizations of the Rosser sentence. But, where Shepherdson merely tacks on side-formulae, Guaspari takes a more revolutionary step: He views the basic components, Pr T (⌈¬ φ ⌉) and Pr T (⌈ φ ⌉), of the Rosser sentence as attempts to refute something and replaces them by refutations of something else. Ignoring their Shepherdsonesque side-formulae (tacked on for the sake of more esoteric applications), our analysis of Guaspari fixed points can be viewed as merely the isolation of those properties whose refutations can be used in this context. In §1 we offer the outcome of this analysis—i.e. a delineation of properties whose refutations can so be used. Some simple examples are cited and the Guaspari fixed points are defined. The main theorem—a master Fixed Point Calculation—is proven in §2.

Some rapidly growing functions

Publisher Summary The purpose of this chapter is pure iconoclasm, the focus being on some rapidly growing functions. When the mathematician says “large,” the logician is sure to think “small.” The first cliche usually resorted to in discussions of largeness is the Skewes number. The Skewes number has toppled from its position of supremacy. In 1955, Skewes showed how to lower the bound if one still assumed the Riemann Hypothesis, but he saved his reputation by obtaining the even larger upper bound. The Ketonen–Solovay elementary proof, like elementary proofs of theorems of analytic number theory, is somewhat longer than the Paris-Harrington proof. However, a bit of the flavor of their proof by showing that H(x + 1, x, x) eventually majorizes all functions F n for finite n. The Ketonen–Solovay elementary proof, like elementary proofs of theorems of analytic number theory, is somewhat longer than the Paris–Harrington proof. The proponents of rapid growth do not stop here, but they seek ever more rapidly growing functions and ever more powerful principles to produce such functions.