## Journal of Symbolic Logic

### The proof-theoretic analysis of transfinitely iterated quasi least fixed points

Dieter Probst

#### Abstract

The starting point of this article is an old question asked by Feferman in his paper on Hancock’s conjecture [6] about the strength of ID₁*. This theory is obtained from the well-known theory ID₁ by restricting fixed point induction to formulas that contain fixed point constants only positively. The techniques used to perform the proof-theoretic analysis of ID₁* also permit to analyze its transfinitely iterated variants IDα*. Thus, we eventually know that |\hat{ID}α| = |IDα*|.

#### Article information

Source
J. Symbolic Logic, Volume 71, Issue 3 (2006), 721-746.

Dates
First available in Project Euclid: 4 August 2006

https://projecteuclid.org/euclid.jsl/1154698573

Digital Object Identifier
doi:10.2178/jsl/1154698573

Mathematical Reviews number (MathSciNet)
MR2250817

Zentralblatt MATH identifier
1115.03084

#### Citation

Probst, Dieter. The proof-theoretic analysis of transfinitely iterated quasi least fixed points. J. Symbolic Logic 71 (2006), no. 3, 721--746. doi:10.2178/jsl/1154698573. https://projecteuclid.org/euclid.jsl/1154698573

#### References

• Peter Aczel, The strength of Martin-Löf's type theory with one universe, Technical report, Department of Philosophy, University of Helsinki, 1977.
• Jeremy Avigad, On the relationship between $\hbox\it ATR_0$ and $\widehat\hbox\it ID_<\omega$, Journal of Symbolic Logic, vol. 61 (1996), no. 3, pp. 768--779.
• Wilfried Buchholz, Solomon Feferman, Wolfram Pohlers, and Wilfried Sieg Iterated inductive definitions and subsystems of analysis: Recent proof-theoretical studies, Lecture Notes in Mathematics, vol. 897, Springer, 1981.,
• Andrea Cantini, A note on a predicatively reducible theory of iterated elementary induction, Bollettino Unione Mathematica Italiana, vol. 4-B (1985), no. 6, pp. 413--430.
• --------, On the relationship between choice and comprehension principles in second order arithmetic, Journal of Symbolic Logic, vol. 51 (1986), pp. 360--373.
• Solomon Feferman, Iterated inductive fixed-point theories: Application to Hancock's conjecture, The Patras Symposion (G. Metakides, editor), North Holland, 1982, pp. 171--196.
• --------, Reflecting on incompleteness, Journal of Symbolic Logic, vol. 56 (1991), no. 1, pp. 1--49.
• Harvey Friedman, Theories of inductive definition, 1969,Unpublished notes.
• Gerhard Jäger, Reinhard Kahle, Anton Setzer, and Thomas Strahm, The proof-theoretic analysis of transfinitely iterated fixed point theories, Journal of Symbolic Logic, vol. 64 (1999), no. 1, pp. 53--67.
• Gerhard Jäger and Dieter Probst, Variation on a theme of Schütte, Mathematical Logic Quarterly, vol. 50 (2004), no. 3, pp. 258--264.
• G. Kreisel, Mathematical Logic, Lectures on modern mathematics (Thomas Lorie Saaty, editor), vol. III, Wiley, 1965, pp. 95--195.
• Wolfram Pohlers Proof Theory: An Introduction, Lecture Notes in Mathematics, vol. 1407, Springer, 1989.,
• Dieter Probst, On the relationship between fixed points and iteration in admissible set theory without foundation, Archive for Mathematical Logic, (2005), no. 44, pp. 561--580.
• -------- Pseudo-hierarchies in admissible set theories without foundation and explicit mathematics, Ph.D. thesis, Universität Bern, 2005.,
• Christian Rüede, The proof-theoretic analysis of $\Sigma_1^1$ transfinite dependent choice, Annals of Pure and Applied Logic, vol. 121 (2003), no. 1, pp. 195--234.
• Kurt Schütte Proof Theory, Springer, 1977.,
• Helmut Schwichtenberg, Proof theory: Some applications of cut-elimination, Handbook of Mathematical Logic (J. Barwise, editor), North Holland, 1977, pp. 867--895.
• Stephen G. Simpson Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic, Springer, 1998.,