Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-12T17:56:38.094Z Has data issue: false hasContentIssue false
  • English
  • Français

DEFINABLE MINIMAL COLLAPSE FUNCTIONS AT ARBITRARY PROJECTIVE LEVELS

Published online by Cambridge University Press:  14 March 2019

VLADIMIR KANOVEI  and
VASSILY LYUBETSKY
VLADIMIR KANOVEI
Affiliation:
IITP RAS, BOLSHOY KARETNY 19 BUILD. 1 MOSCOW 127051, RUSSIAE-mail: kanovei@googlemail.com
VASSILY LYUBETSKY
Affiliation:
IITP RAS, BOLSHOY KARETNY 19 BUILD. 1 MOSCOW 127051, RUSSIAE-mail: lyubetsk@iitp.ru
Get access Rights & Permissions [Opens in a new window]

Abstract

Using a nonLaver modification of Uri Abraham’s minimal $\Delta _3^1$ collapse function, we define a generic extension $L[a]$ by a real a, in which, for a given $n \ge 3$, $\left\{ a \right\}$ is a lightface $\Pi _n^1 $ singleton, a effectively codes a cofinal map $\omega \to \omega _1^L $ minimal over L, while every $\Sigma _n^1 $ set $X \subseteq \omega $ is still constructible.

Keywords

03E3503E4503E15forcingminimal collapse functionsprojective classes
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 1 , March 2019 , pp. 266 - 289
Copyright
Copyright © The Association for Symbolic Logic 2019 

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.)

References

REFERENCES

Abraham, U., A minimal model for $\neg \,CH:$ Iteration of Jensen’s reals. Transactions of the American Mathematical Society , vol. 281 (1984), pp. 657–674.Google Scholar
Abraham, U., Minimal model of “$\aleph _1^L $ is countable” and definable reals. Advances in Mathematics, vol. 55 (1985), pp. 75–89.Google Scholar
Barwise, J. (ed.), Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, vol. 90, North-Holland, Amsterdam, 1977.10.1016/S0049-237X(08)71097-8CrossRefGoogle Scholar
Bukovsky, L., Changing cofinality of $\aleph _2 $, Set Theory and Hierarchy Theory, Memorial Tribute A. Mostowski (Marek, W., Srebrny, M., and Zarach, A., editors), Lecture Notes in Mathematics, vol. 537, Springer-Verlag, Berlin, 1976, pp. 37–49.10.1007/BFb0096892CrossRefGoogle Scholar
Bukovský, L. and Copláková-Hartová, E., Minimal collapsing extensions of models of ZFC. Annals of Pure and Applied Logic, vol. 46 (1990), no. 3, pp. 265–298.Google Scholar
Cummings, J., Iterated forcing and elementary embeddings, Handbook of Set Theory, vol. 3 (Foreman, M. and Kanamori, A., editors), Springer, Dordrecht, 2010, pp. 775–883.10.1007/978-1-4020-5764-9_13CrossRefGoogle Scholar
Fischer, V., Friedman, S. D., Mejía, D. A., and Montoya, D. C., Coherent systems of finite support iterations, this Journal, vol. 83 (2018), no. 1, pp. 208–236.Google Scholar
Friedman, S.-D., Gitman, V., and Kanovei, V., A model of second-order arithmetic satisfying AC but not DC. Journal of Mathematical Logic, to appear, 2018, arXiv:1808.04732.Google Scholar
Golshani, M., Kanovei, V., and Lyubetsky, V., A Groszek – Laver pair of undistinguishable $E_0 $ classes . Mathematical Logic Quarterly, vol. 63 (2017), no. 1–2, pp. 19–31.Google Scholar
Harrington, L., Handwritten notes, in four parts: (A) The constructible reals can be anything. May 1974.(B) Addendum. Models where Separation principles fail. May 1974. (C) Separation without Reduction. April 1975. (D) The constructible reals can be (almost) anything. Part II. May 1975.Google Scholar
Jech, T., Set Theory, The third millennium revised and expanded edition, Springer-Verlag, Berlin-Heidelberg, New York, 2003.Google Scholar
Jensen, R. B. and Solovay, R. M., Some applications of almost disjoint sets, Mathematical Logic and Foundations of Set Theory Proceedings of an International Colloquium, Jerusalem 1968 (Bar-Hillel, Y., editor), North-Holland, Amsterdam-London, 1970, pp. 84–104.Google Scholar
Jensen, R., Definable sets of minimal degree, Mathematical Logic and Foundations of Set Theory Proceedings of an International Colloquium, Jerusalem 1968 (Bar-Hillel, Y., editor), North-Holland, Amsterdam-London, 1970, pp. 122–128.Google Scholar
Kanamori, A., Perfect-set forcing for uncountable cardinals. Annals of Mathematical Logic, vol. 19 (1980), pp. 97–114.Google Scholar
Kanovei, V., On the nonemptiness of classes in axiomatic set theory. Mathematics of the USSR-Izvestiya, vol. 12 (1978), pp. 507–535.Google Scholar
Kanovei, V., Non-Glimm-Effros equivalence relations at second projective level. Fundamenta Mathematicae, vol. 154 (1997), no. 1, pp. 1–35.Google Scholar
Kanovei, V. and Lyubetsky, V., A definable $E_0 $-class containing no definable elements. Archive for Mathematical Logic, vol. 54 (2015), no. 5, pp. 711–723.Google Scholar
Kanovei, V., Counterexamples to countable-section ${\rm{\Pi }}_2^1 $ uniformization and ${\rm{\Pi }}_3^1 $ separation. Annals of Pure and Applied Logic, vol. 167 (2016), no. 4, pp. 262–283.Google Scholar
Kanovei, V., Definable $E_0 $ classes at arbitrary projective levels. Annals of Pure and Applied Logic, vol. 169 (2018), no. 9, pp. 851–871.Google Scholar
Kanovei, V., Non-uniformizable sets of second projective level with countable cross-sections in the form of Vitali classes. Izvestiya: Mathematics, vol. 82 (2018), no. 1, pp. 65–96.Google Scholar
Kanovei, V., Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy. Fundamenta Mathematicae, to appear, (2018).Google Scholar
Kurilić, M. S., Changing cofinalities and collapsing cardinals in models of set theory. Annals of Pure and Applied Logic, vol. 120 (2003), no. 1–3, pp. 225–236.Google Scholar
Namba, K., Independence proof of $\left( {\omega ,\omega _\alpha } \right)$-distributive law in complete Boolean algebras. Commentarii Mathematici Universitatis Sancti Pauli, vol. 19 (1971), pp. 1–12.Google Scholar