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A Model-Theoretic Approach to Ordinal Analysis

  • Jeremy Avigad (a1) and Richard Sommer (a2)

We describe a model-theoretic approach to ordinal analysis via the finite combinatorial notion of an α-large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first- and second-order arithmetic.

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[2] Jeremy Avigad , Formalizing forcing arguments in subsystems of second-order arithmetic, Annals of Pure and Applied Logic, vol. 82 (1996), pp. 165191.

[5] Jon Barwise , The handbook of mathematical logic, North-Holland, 1977.

[7] Solomon Feferman , Iterated inductive fixed-point theories: Application to Hancock's conjecture, Patras logic symposium ( G. Metakides , editor), North-Holland, 1982.

[9] Harvey Friedman , Kenneth McAloon , and Stephen Simpson , A finite combinatorial principle which is equivalent to the 1-consistency of predicative analysis, Patras logic symposium ( G. Metakides , editor), North-Holland, 1982, pp. 197230.

[12] Jussi Ketonen and Robert Solovay , Rapidly growing Ramsey functions, Annals of Mathematics, vol. 113 (1981), pp. 267314.

[14] H. Kotlarski and Z. Ratajczyk , Inductive full satisfaction classes, Annals of Pure and Applied Logic, vol. 47 (1990), pp. 199223.

[15] Jeff B. Paris , A hierarchy of cuts in models of arithmetic, Springer-Verlag Lecture Notes in Mathematics, vol. 834, 1980, pp. 312337.

[17] Wolfram Pohlers , Proof theory: An introduction, Springer-Verlag Lecture Notes in Mathematics, vol. 1407, 1989.

[18] Wolfram Pohlers , A short course in ordinal analysis, Proof theory ( Aczel et al., editors), Cambridge University Press, 1993.

[19] Zygmunt Ratatczyk , Subsystems of true arithmetic and hierarchies of functions, Annals of Pure and Applied Logic, vol. 64 (1993), pp. 95152.

[21] Michael Rathjen , Proof theory of reflection, Annals of Pure and Applied Logic, vol. 68 (1994), pp. 181224.

[24] Kurt Schütte , Proof theory, Springer-Verlag, 1977.

[29] Rick Smith , The consistency strengths of some finite forms of the Higman and Kruskal theorems, Harvey Friedman's research on the foundations of mathematics ( Leo Harrington et al., editors), North-Holland, 1985, pp. 119135.

[32] Richard Sommer , Transfinite induction within Peano arithmetic, Annals of Pure and Applied Logic, vol. 76 (1995), pp. 231289.

[33] S. S. Wainer , A classification of the ordinal recursive functions, Archiv für mathematische Logik, vol. 13 (1970), pp. 136153.

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Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
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