Hostname: page-component-cb9f654ff-c75p9 Total loading time: 0 Render date: 2025-09-01T03:46:14.111Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE CATEGORICITY OF COMPLETE SECOND-ORDER THEORIES

Part of: Set theory Model theory

Published online by Cambridge University Press:  27 August 2025

TAPIO SAARINEN Open the ORCID record for TAPIO SAARINEN [Opens in a new window] ,
JOUKO ANTERO VÄÄNÄNEN Open the ORCID record for JOUKO ANTERO VÄÄNÄNEN [Opens in a new window]  and
WILLIAM HUGH WOODIN
TAPIO SAARINEN
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF HELSINKI HELSINKI, FINLAND E-mail: tapio.saarinen@helsinki.fi
JOUKO ANTERO VÄÄNÄNEN*
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF HELSINKI HELSINKI, FINLAND AND ILLC UNIVERSITY OF AMSTERDAM AMSTERDAM, NETHERLANDS
WILLIAM HUGH WOODIN
Affiliation:
DEPARTMENT OF MATHEMATICS DEPARTMENT OF PHILOSOPHY HARVARD UNIVERSITY CAMBRIDGE, MA 02138, USA E-mail: wwoodin@g.harvard.edu
*
E-mail: jouko.vaananen@helsinki.fi
Get access Rights & Permissions [Opens in a new window]

Abstract

We show, assuming PD, that every complete finitely axiomatized second-order theory with a countable model is categorical, but that there is, assuming again PD, a complete recursively axiomatized second-order theory with a countable model which is non-categorical. We show that the existence of even very large (e.g., supercompact) cardinals does not imply the categoricity of all finitely axiomatizable complete second-order theories. More exactly, we show that a non-categorical complete finitely axiomatized second-order theory can always be obtained by (set) forcing. We also show that the categoricity of all finite complete second-order theories with a model of a certain singular cardinality $\kappa $ of uncountable cofinality can be forced over any model of set theory. Previously, Solovay had proved, assuming $V=L$, that every complete finitely axiomatized second-order theory (with or without a countable model) is categorical, and that in a generic extension of L there is a complete finitely axiomatized second-order theory with a countable model which is non-categorical.

Keywords

categoricitysecond-order logicprojective determinacyforcing

MSC classification

Primary: 03C55: Set-theoretic model theory 03C85: Second- and higher-order model theory 03E35: Consistency and independence results 03E47: Other notions of set-theoretic definability

Information

Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 23
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

Ajtai, M., Isomorphism and higher order equivalence . Annals of Mathematical Logic, vol. 16 (1979), no. 3, pp. 181–203.10.1016/0003-4843(79)90001-9CrossRefGoogle Scholar
Asperó, D. and Friedman, S.-D., Large cardinals and locally defined well-orders of the universe . Annals of Pure and Applied Logic, vol. 157 (2009), no. 1, pp. 1–15.10.1016/j.apal.2008.07.006CrossRefGoogle Scholar
Asperó, D. and Friedman, S.-D., Definable well-orders of $H\left({\omega}_2\right)$ and GCH. Journal of Symbolic Logic, vol. 77 (2012), no. 4, pp. 1101–1121.10.2178/jsl.7704030CrossRefGoogle Scholar
Asperó, D. and Schindler, R., Martin’s Maximum ${}^{++}$ implies Woodin’s axiom $\left(\ast \right)$ . Annals of Mathematics (2), vol. 193 (2021), no. 3, pp. 793–835.10.4007/annals.2021.193.3.3CrossRefGoogle Scholar
Awodey, S. and Reck, E. H., Completeness and categoricity. I. Nineteenth-century axiomatics to twentieth-century metalogic . History and Philosophy of Logic, vol. 23 (2002), no. 1, pp. 1–30.10.1080/01445340210146889CrossRefGoogle Scholar
Carnap, R., Untersuchungen zur Allgemeinen Axiomatik, Edited and with a foreword by (Bonk, T. and Mosterin, J., editors), Wissenschaftliche Buchgesellschaft, Darmstadt, 2000.Google Scholar
Conway, J. H., Homogeneous Ordered Sets. Ph.D. thesis, Cambridge, England, 1964.Google Scholar
Cummings, J., Friedman, S.-D., Magidor, M., Sinapova, D., and Rinot, A., Ordinal definable subsets of singular cardinals . Israel Journal of Mathematics, vol. 226 (2018), no. 2, pp. 781–804.10.1007/s11856-018-1712-2CrossRefGoogle Scholar
Fraenkel, A., Einleitung in die Mengenlehre, 3rd ed., Springer, Berlin, 1928.10.1007/978-3-662-42029-4CrossRefGoogle Scholar
Fraïssé, R., Sur les types de polyrelations et sur une hypothèse d’origine logistique . Comptes Rendus Mathématique. Académie des Sciences. Paris, vol. 230 (1950), pp. 1557–1559.Google Scholar
Fraïssé, R., Sur la signification d’une hypothèse de la théorie des relations, du point de vue du calcul logique . Comptes Rendus Mathématique. Académie des Sciences. Paris, vol. 232 (1951), pp. 1793–1795.Google Scholar
Kanamori, A., The Higher Infinite: Large Cardinals in Set Theory from their Beginnings, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2009.Google Scholar
Kaplan, I. and Shelah, S., Forcing a countable structure to belong to the ground model . Mathematical Logic Quarterly, vol. 62 (2016) no. 6, pp. 530–546.10.1002/malq.201400094CrossRefGoogle Scholar
Kleinberg, E. M., Infinitary Combinatorics and the Axiom of Determinateness, Lecture Notes in Mathematics, 612, Springer-Verlag, Berlin, 1977.10.1007/BFb0069385CrossRefGoogle Scholar
Koellner, P. and Woodin, W. H., Large cardinals from determinacy , Handbook of Set Theory (Foreman, M. and Kanamori, A., editors), Springer, Dordrecht, 2010, pp. 1951–2119.10.1007/978-1-4020-5764-9_24CrossRefGoogle Scholar
Kunen, K., Elementary embeddings and infinitary combinatorics . Journal of Symbolic Logic, vol. 36 (1971), pp. 407–413.10.2307/2269948CrossRefGoogle Scholar
Laskowski, M. C. and Shelah, S., On the existence of atomic models . Journal of Symbolic Logic, vol. 58 (1993), no. 4, pp. 1189–1194.10.2307/2275137CrossRefGoogle Scholar
Laver, R., Certain very large cardinals are not created in small forcing extensions . Annals of Pure and Applied Logic, vol. 149 (2007), nos. 1–3, pp. 1–6.10.1016/j.apal.2007.07.002CrossRefGoogle Scholar
Louveau, A., Borel sets and the analytical hierarchy , Proc. Herbrand Symp. Logic Colloquium ’81 (Stern, J., editor), North-Holland, Amsterdam, 1982, pp 209–215.10.1016/S0049-237X(08)71885-8CrossRefGoogle Scholar
Marek, W., Consistance d’une hypothèse de Fraïssé sur la définissabilité dans un langage du second ordre . Comptes Rendus Mathématique. Académie des Sciences Séries A-B, vol. 276 (1973), pp. A1147–A1150.Google Scholar
Marek, W., Sur la consistance d’une hypothèse de Fraïssé sur la définissabilité dans un langage du second ordre . Comptes Rendus Mathématique. Académie des Sciences Séries A-B, vol. 276 (1973), pp. A1169–A1172.Google Scholar
Martin, D. A., The axiom of determinateness and reduction principles in the analytical hierarchy . Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687–689.10.1090/S0002-9904-1968-11995-0CrossRefGoogle Scholar
Menas, T. K., Consistency results concerning supercompactness . Transactions of the American Mathematical Society, vol. 223 (1976), pp. 61–91.10.1090/S0002-9947-1976-0540771-8CrossRefGoogle Scholar
Moschovakis, Y. N., Descriptive Set Theory, 2nd ed., Mathematical Surveys and Monographs, 155, American Mathematical Society, Providence, 2009.10.1090/surv/155CrossRefGoogle Scholar
Nadel, M. and Stavi, J., ${L}_{\infty \lambda }$ -equivalence, isomorphism and potential isomorphism . Transactions of the American Mathematical Society, vol. 236 (1978), pp. 51–74.Google Scholar
Rabin, M. O., A simple method for undecidability proofs and some applications , Logic, Methodology and Philosophy of Science (Proceedings of the 1964 International Congress), North-Holland, Amsterdam, 1965, pp 58–68.Google Scholar
Rudominer, M., The mouse set theorem just past projective . Journal of Mathematical Logic, vol. 25 (2025), article no. 2450014 (32 pp). https://doi.org/10.1142/S0219061324500144 CrossRefGoogle Scholar
Shelah, S., Set theory without choice: Not everything on cofinality is possible . Archive for Mathematical Logic, vol. 36 (1997), no. 2, pp. 81–125.10.1007/s001530050057CrossRefGoogle Scholar
Solovay, R., FOM posting, 2006. Available at http://cs.nyu.edu/pipermail/fom/2006-May/010561.html.Google Scholar
Steel, J. R., Scales in L(R) , Cabal Seminar 79–81, Lecture Notes in Mathematics, 1019, Springer, Berlin, 1983, pp. 107–156.10.1007/BFb0071699CrossRefGoogle Scholar
Woodin, W. H., The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, volume 1 of De Gruyter Series in Logic and its Applications, Walter de Gruyter GmbH & Co. KG, Berlin, 2010. revised edition.10.1515/9783110213171CrossRefGoogle Scholar
Woodin, W. H., Suitable extender models II: Beyond $\omega$ -huge . Journal of Mathematical Logic, vol. 11 (2011), no. 2, pp. 115–436.10.1142/S021906131100102XCrossRefGoogle Scholar
Woodin, W. H., In search of ultimate- $L$ : The 19th midrasha mathematicae lectures . Bulletin of Symbolic Logic, vol. 23 (2017), no. 1, pp. 1–109.10.1017/bsl.2016.34CrossRefGoogle Scholar