Skip to main content
Log in

Arithmetical algorithms for elementary patterns

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

Elementary patterns of resemblance notate ordinals up to the ordinal of \({\Pi^1_1 - CA_0}\). We provide ordinal multiplication and exponentiation algorithms using these notations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alexander, S.: The patterns of resemblance ordinal calculator. http://www.xamuel.com/patterns/ (2012)

  2. Bès A.: Decidability and definability results related to the elementary theory of ordinal multiplication. Fundamenta Mathematicae 171, 197–211 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Carlson T.J.: Ordinal arithmetic and Σ1 elementarity. Arch. Math. Logic 38, 449–460 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Carlson T.J.: Knowledge, machines, and the consistency of Reinhardt’s strong mechanistic thesis. Ann. Pure Appl. Logic 105, 51–82 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Carlson T.J.: Elementary patterns of resemblance. Ann. Pure Appl. Logic 108(1), 19–77 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Carlson T.J.: Patterns of resemblance of order 2. Ann. Pure Appl. Logic 158(1–2), 90–124 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Carlson T.J., Wilken G.: Normal forms for elementary patterns. J. Symb. Logic 77(1), 174–194 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Wilken, G.: Σ1-elementarity and Skolem hull operators. Thesis (dissertation), University of Münster (2004)

  9. Wilken G.: The Bachmann–Howard structure in terms of Σ1-elementarity. Arch. Math. Logic 45(7), 807–829 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Wilken G.: Assignment of ordinals to elementary patterns of resemblance. J. Symb. Logic 72(2), 704–720 (2007)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Samuel A. Alexander.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alexander, S.A. Arithmetical algorithms for elementary patterns. Arch. Math. Logic 54, 113–132 (2015). https://doi.org/10.1007/s00153-014-0404-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-014-0404-9

Mathematics Subject Classification

Navigation