Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-11T16:52:43.898Z Has data issue: false hasContentIssue false
  • English
  • Français

On the relationship between ATR0 and

Published online by Cambridge University Press:  12 March 2014

Jeremy Avigad
Jeremy Avigad*
Affiliation:
Department of Philosophy, Carnegie Mellon University, Pittsburgh, PA 15213, USA, E-mail: avigad@cmu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the theory ATR0 is equivalent to a second-order generalization of the theory . As a result, ATR0 is conservative over for arithmetic sentences, though proofs in ATR0 can be much shorter than their counterparts.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 3 , September 1996 , pp. 768 - 779
Copyright
Copyright © Association for Symbolic Logic 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Avigad, Jeremy, Proof-theoretic investigations of subsystems of second-order arithmetic, Ph.d. dissertation, U. C. Berkeley, 1995.Google Scholar
[2]Feferman, Solomon, Iterated inductive fixed-point theories: Application to Hancock's conjecture, Patras logic symposium (Metakides, G., editor), North-Holland Publishing Company, 1982.Google Scholar
[3]Friedman, Harvey, McAloon, Kenneth, and Simpson, Stephen, A finite combinatorial principle which is equivalent to the 1-consistency of predicative analysis, Patras logic symposium (Metakides, G., editor), North-Holland Publishing Company, 1982.Google Scholar
[4]Hájek, Petr and Pudlak, Pavel, Metamathematics of first-order arithmetic, Springer-Verlag, 1991.Google Scholar
[5]Ignjatovic, Aleksandar, Fragments of first and second order arithmetic and length of proofs, Ph.d. dissertation, U. C. Berkeley, 1990.Google Scholar
[6]Jäger, Gerhard, Theories for admissible sets: A unifying approach to proof theory, Bibliopolis, 1986.Google Scholar
[7]Jech, Thomas, Set theory, Academic Press, 1978.Google Scholar
[8]Martin, Donald, Descriptive set theory: Projective sets, The handbook of mathematical logic (Barwise, Jon, editor), North-Holland, 1991.Google Scholar
[9]Pohlers, Wolfram, Proof theory: An introduction, Lecture notes in mathematics 1407, Springer Verlag, 1989.Google Scholar
[10]Pudlák, Pavel, Cuts, consistency statements, and interpretations, this Journal, vol. 50 (1985), pp. 423–441.Google Scholar
[11]Pudlák, Pavel, On the length of proofs of finitistic consistency statements in first order theories, Logic Colloqium '84 (Paris, et al., editors), North-Holland, 1986.Google Scholar
[12]Schütte, Kurt, Proof theory, vol. 84, Springer-Verlag, 1977.CrossRefGoogle Scholar
[13]Schwichtenbero, Helmut, Proof theory: Some applications of cut-elimination, The handbook of mathematical logic (Barwise, Jon, editor), North-Holland, 1991.Google Scholar
[14]Sieg, Wilfried, Herbrand analysis, Archive for Mathematical Logic, vol. 30 (1991), pp. 409–441.CrossRefGoogle Scholar
[15]Simpson, Stephen G., Subsystems of second order arithmetic, preprint.Google Scholar
[16]Simpson, Stephen G., and transfinite induction, Logic Colloqium (van Dalen, D.et al., editors), vol. 80, North-Holland, 1982.Google Scholar
[17]Simpson, Stephen G., Subsystems of z2 and reverse mathematics, appendix to Gaisi Takeuti, Proof Theory, second edition, North-Holland, 1987.Google Scholar
[18]Simpson, Stephen G., On the strength of König's duality theorem of countable bipartite graphs, this Journal, vol. 59 (1994), pp. 113–123.Google Scholar