The Wonder of Colors and the Principle of Ariadne

  • Walter CarnielliEmail author
  • Carlos di Prisco
Part of the Synthese Library book series (SYLI, volume 388)


This paper surveys some results on combinatorial aspects of infinite Ramsey-type problems inspired by finite properties, and intends to explain the relevance of an alternative set-theoretical principle formulated in the language of colors, the so-called Principle of Ariadne. This principle, intended to be a rival of the Axiom of Choice, can be consistently added to the usual axiom stock of ZF set theory under certain conditions. Such a new axiom, which preserves all the finite contents of mathematics but deviates from the standard in the infinite contents, may help us to understand the finite-infinite divide in mathematics, making clear that there is more than one way to generalize from finite principles of order (or choice) to the infinite. In other words, several infinite principles are possible starting from the same finitary content.



Some results in this paper have been written years ago during a visit of the first named author to the Instituto Venezolano de Investigaciones Científicas (IVIC) in Caracas, Venezuela. This author also acknowledges support from FAPESP Thematic Project LogCons 2010/51038-0, Brazil, and from a research grant from the National Council for Scientific and Technological Development (CNPq), Brazil.


  1. Carnielli, W. A. (1986). Ramsey-type theorems for cubes. In A. M. Gleason (Ed.), Abstracts of the International Congress of Mathematicians ’86, Berkeley (p. 305). American Mathematical Society.Google Scholar
  2. Carnielli, W. A. (1996). Auto-organização em estruturas combinatórias. In O. Pessoa, Jr., M. M. Debrun, & M. E. Q. Gonzalez (Eds.), Auto-Organização: Estudos Interdisciplinares (Vol. 18, pp. XX–YY). Coleção CLE, CLE- Unicamp.Google Scholar
  3. Carnielli, W. A. & Di Prisco, C. A. (1988). Polarized partition relations for higher dimension (Vol. 12). Technical report, Department of Mathematics, University of Campinas. Published as Relatório Técnico do Instituto de Matemática da Universidade Estadual de Campinas.Google Scholar
  4. Carnielli, W. A. & Di Prisco, C. A. (1993) Some results on polarized partition relations of higher dimension. Mathematical Logic Quarterly, 39, 461–474.CrossRefGoogle Scholar
  5. Carrasco, M., Di Prisco, C. A., & Millán, A. (1995). Partitions of the set of finite sequences. Journal of Combinatorial Theory, Series A, 71(2), 255–274.CrossRefGoogle Scholar
  6. Erdős, P. & Rado, R. (1956). A partition calculus in set theory. Bulletin of the American Mathematical Society, 62(5), 427–489.Google Scholar
  7. Erdős, P., Hajnal, A., & Rado, R. (1965). Partition relations for cardinal numbers. Acta Mathematica Academiae Scientiarum Hungarica, 16, 93–196.Google Scholar
  8. Feferman, S., Friedman, H., Maddy, P., & Steel, J. (2000). Does mathematics need new axioms? The Bulletin of Symbolic Logic, 6, 401–446.CrossRefGoogle Scholar
  9. Lolli, G. (1977). On Ramsey’s theorem and the Axiom of Choice. Notre Dame Journal of Formal Logic, 18(4), 599–601.CrossRefGoogle Scholar
  10. Mathias, A. R. D. (1977). Happy families. Annals of Mathematical Logic, 12(1), 59–111.CrossRefGoogle Scholar
  11. Mycielski, J. & Steinhaus, H. (1962). A mathematical axiom contradicting the axiom of choice. Bulletin de Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, 10, 1–3.Google Scholar
  12. Paris, J. & Harrington, L. A. (1977). A mathematical incompleteness in Peano Arithmetic. In J. Barwise (Ed.), Handbook of mathematical logic (pp. 1133–1142). Amsterdam: North-Holland.CrossRefGoogle Scholar
  13. Patey, L. & Yokoyama, K. (2016). The proof-theoretic strength of Ramsey’s theorem for pairs and two colors. ArXiv e-printsGoogle Scholar
  14. Di Prisco, C. A. & Henle, J. M. (1993). Partitions of products. Journal of Symbolic Logic, 58(3), 860–871.CrossRefGoogle Scholar
  15. Di Prisco, C. A., & Henle, J. (1999). Partitions of the reals and choice. In X. Caicedo, & C. H. Montenegro (Eds.), Selected Papers of the X Latin American Symposium on Mathematical Logic (pp. 13–23). Marcel Dekker, Inc.Google Scholar
  16. Di Prisco, C. A. & Todorcevic, S. (2003). Souslin partitions of products of finite sets. Advances in Mathematics, 176, 145–173.CrossRefGoogle Scholar
  17. Quine, W. V. O. (1969). Natural kinds. In J. Kim & E. Sosa (Eds.), Ontological relativity and other essays (pp. 114–138). New York: Columbia University Press.Google Scholar
  18. Ramsey, F. P. (1930). On a problem of formal logic. Proceedings of the London Mathematical Society, 30, 264–286.Google Scholar
  19. Shelah, S. (1998). A polarized partition relation and failure of GCH at singular strong limit. Fundamenta Mathematicae, 155(2), 153–160.Google Scholar
  20. Wittgenstein, L. (1998). Culture and value (P. Winch, Trans.). Basil Blackwell. Organized by G. H. von Wright in collaboration with H. Nyman.Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of PhilosophyCentre for Logic, Epistemology and the History of Science, State University of Campinas UNICAMPCampinasBrazil
  2. 2.Instituto Venezolano de Investigaciones CientíficasUniversidad de Los AndesBogotáColombia

Personalised recommendations