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A stronger version of Friedman’s self-embedding theorem

Keita Yokoyama

May 20, 2011

In [1], Harvey Friedman showed the famous self-embedding theorem for PA which asserts that every countable model of PA has an initial segment which is isomorphic to itself. This theorem can be generalize to the following (see e.g., [2, Section 12]):

(†) every countable model of IΣn has a Σn-elementary initial segment which is isomorphic to itself.

However, this theorem is not strong enough to characterize countable models of IΣn, i.e., there exists a countable model M which satisfies (†) but M 6|= IΣn. On the other hand, Tanaka’s self- embedding theorem for WKL0[3] exactly characterize countable models of WKL0. We consider a stronger version of (†) and characterize models of subsystems of PA.

References

[1] Harvey Friedman. Countable models of set theories. In Cambridge Summer School in Math. Logic, volume 337 of Lecture Notes in Math., pages 539–573, 1973.

[2] Richard Kaye. Models of Peano Arithmetic. Oxford Logic Guides, 15. Oxford University Press, 1991. x+292 pages.

[3] Kazuyuki Tanaka. The self-embedding theorem of WKL0and a non-standard method. Annals of Pure and Applied Logic, 84:41–49, 1997.

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