Archive for Mathematical Logic

, Volume 56, Issue 5–6, pp 699–711 | Cite as

The strength of countable saturation

  • Benno van den BergEmail author
  • Eyvind Briseid
  • Pavol Safarik
Open Access


In earlier work we introduced two systems for nonstandard analysis, one based on classical and one based on intuitionistic logic; these systems were conservative extensions of first-order Peano and Heyting arithmetic, respectively. In this paper we study how adding the principle of countable saturation to these systems affects their proof-theoretic strength. We will show that adding countable saturation to our intuitionistic system does not increase its proof-theoretic strength, while adding it to the classical system increases the strength from first- to full second-order arithmetic.


Proof theory Nonstandard analysis Arithmetic in all finite types Saturation principles 

Mathematics Subject Classification

03F10 03F25 03F50 11U10 26E35 


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Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute for Logic, Language and ComputationUniversiteit van AmsterdamAmsterdamThe Netherlands
  2. 2.GFU/LUIOslo and Akershus University College of Applied SciencesOsloNorway
  3. 3.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany

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