Who Put The “Back” In Back-And-Forth?

  • J. M. Plotkin
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 12)


Logicians of a certain age can remember when they were first allowed to savor Cantor’s sweet theorem characterizing the rationals among ordered sets as the unique countable densely ordered set without endpoints. For most the pleasure was heightened when they were shown the elegant back-and-forth argument that makes the proof of Cantor’s theorem perfectly transparent. This method of establishing isomorphism is now so widely known that its invocation no longer begins demonstrations, it ends them. Provers tell us: “By the usual back-and-forth argument so-and-so is isomorphic to such-and-such”, and then they move on to other matters. Logicians, in particular model-theorists, deserve a great deal of credit for making back-and-forth a mathematical cliché.


Order Relationship Countable Structure Sixtieth Birthday Partial Isomorphism Orth Argument 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Cameron, P.J. [1990], Oligomorphic Permutation Groups. London Math. Society Lecture Notes Series, vol. 152, Cambridge University Press.Google Scholar
  2. Cantor, G. [1895], Beiträge zur Begründung der transfiniten Mengenlehre I. Mathematische Annalen, 46, 481–512.zbMATHCrossRefGoogle Scholar
  3. Cantor, G. [1915], Contributions To The Founding Of The Theory Of Transfinite Numbers. English translation by Ph.E.B. Jourdain. Dover, New York (1952).zbMATHGoogle Scholar
  4. Hausdorff, F. [1907], Untersuchungen über Ordnungstypen. Berichte über die Verhandlungen der Sachsischen Gesellschaft der Wissenschaften zu Leipzig, 59, 84–159.Google Scholar
  5. Hausdorff, F. [1908], Grundzüge einer Theorie der geordneten Mengen. Mathematische Annalen, 65, 435–505.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Hausdorff, F. [1914], Grundzuge der Mengenlehre. Veit, Leipzig; reprinted, Chelsea, New York (1949).Google Scholar
  7. Hausdorff, F. [1927], Mengenlehre. Walter de Gruyter, Berlin and Leipzig.zbMATHGoogle Scholar
  8. Hodges, W. [1985], Building Models by Games. London Math. Society Student Texts, 2, Cambridge University Press.Google Scholar
  9. Huntington, E.V. [1905], The continuum as a type of order: an exposition of the modern theory. Annals of Mathematics, 6, 151–184.MathSciNetzbMATHCrossRefGoogle Scholar
  10. Kueker, D.W. [1975], Back-And-Forth Arguments and Infïnitary Logics. In: Infinitary Logic: In Memoriam Carol Karp. D.W. Kueker (ed.), Springer-Verlag, Lecture Notes in Mathematics, 492, 17–71.Google Scholar
  11. Skolem, Th. [1970], Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit und Beweisbarkeit mathematischen Sätze nebst einem Theoreme über dichte Mengen. Selected Works in Logic by Th. Skolem, J.E. Fenstad (ed.), Universitetsforlaget, Oslo, 103–136.Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • J. M. Plotkin
    • 1
  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA

Personalised recommendations