My route to arithmetization

Theoria 63 (3):168-181 (1997)
I had the pleasure of renewing my acquaintance with Per Lindström at the meeting of the Seventh Scandinavian Logic Symposium, held in Uppsala in August 1996. There at lunch one day, Per said he had long been curious about the development of some of the ideas in my paper [1960] on the arithmetization of metamathematics. In particular, I had used the construction of a non-standard definition !* of the set of axioms of P (Peano Arithmetic) to show that P + {¬ Con!} is interpretable in P, where ! is a standard definition of the axioms of P and Con! expresses the consistency of P via that presentation. Per pointed out that there is a simple “two-line” proof of this interpretability result which does not require the use of such formulas as !*, and he wondered whether I had been aware of that. In fact, his proof had never occurred to me, and if it had at the time, it is possible I would never have been led to the use of non-natural definitions. Per regarded this as a happy accident, since subsequent work by him and others on interpretability made essential use of such definitions. In our conversation, I enlarged a bit on the background to my work on arithmetization, and when I was invited to contribute to this special issue of Theoria dedicated to Lindström, it seemed a natural choice to use the occasion to fill out the story. One caveat, though: the following is drawn largely from memory, not always reliable, supplemented by consultation of the 1960 paper and the 1957 dissertation from which it was drawn.
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DOI 10.1111/j.1755-2567.1997.tb00746.x
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References found in this work BETA
The Incompleteness Theorems.Craig Smorynski - 1977 - In Jon Barwise (ed.), Handbook of Mathematical Logic. North-Holland. pp. 821 -- 865.
On Axiomatizability Within a System.William Craig - 1953 - Journal of Symbolic Logic 18 (1):30-32.
Reflection Principles and Their Use for Establishing the Complexity of Axiomatic Systems.Georg Kreisel & Azriel Lévy - 1968 - Zeitschrift für Mathematische Logic Und Grundlagen der Mathematik 14 (1):97--142.

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Resplendent Models and {Sigma_1^1} -Definability with an Oracle.Andrey Bovykin - 2008 - Archive for Mathematical Logic 47 (6):607-623.
The Predicative Frege Hierarchy.Albert Visser - 2009 - Annals of Pure and Applied Logic 160 (2):129-153.
A History of Theoria.Sven Ove Hansson - 2009 - Theoria 75 (1):2-27.

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