It is Sunday, the 7th of September 1930. The place is Konigsberg and the occasion is a small conference on the foundations of mathematics. Arend Heyting, the foremost disciple of L. E. J. Brouwer, has spoken on intuitionism; Rudolf Carnap of the Vienna Circle has expounded on logicism; Johann (formerly Janos and in a few years to be Johnny) von Neumann has explained Hilbert's proof theory-- the so-called formalism; and Hans Hahn has just propounded his own empiricist views of mathematics. The floor is open for general discussion, in the midst of which Heyting announces his satisfaction with the meeting. For him, the relationship between formalism and intuitionism has been clarified: There need be no war between the intuitionist and the formalist. Once the formalist has successfully completed Hilbert's programme and shown "finitely" that the "idealised" mathematics objected to by Brouwer proves no new "meaningful" statements, even the intuitionist will fondly embrace the infinite. To this euphoric revelation, a shy young man cautions~ "According to the formalist conception one adjoins to the meaningful statements of mathematics transfinite (pseudo-')statements which in themselves have no meaning but only serve to make the system a well-rounded one just as in geometry one achieves a well� rounded system by the introduction of points at infinity.

Contenuti

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. Solovay’s First Completeness Theorem.- 2. Solovay’s 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. Carlson’s 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. Švejdar’s Self-Referential Formulae.

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