Skip to main content
SpringerLink
Log in

ZF + DC + AX4

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

We consider mainly the following version of set theory: “ZF+DC and for every \({\lambda, \lambda^{\aleph_0}}\) is well ordered”, our thesis is that this is a reasonable set theory, e.g. on the one hand it is much weaker than full choice, and on the other hand much can be said or at least this is what the present work tries to indicate. In particular, we prove that for a sequence \({\overline{\delta} = \langle\delta_{s}: s \in Y\rangle, {\rm cf}(\delta_{s})}\) large enough compared to Y, we can prove the pcf theorem with minor changes (in particular, using true cofinalities not the pseudo ones).We then deduce the existence of covering numbers and define and prove existence of a class of true successor cardinals. Using this we give some diagonalization arguments (more specifically some black boxes and consequences) on Abelian groups, chosen as a characteristic case.We end by showing that some such consequences hold even in ZF above.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Eklof, P.C., Mekler, A.: Almost Free Modules: Set Theoretic Methods, North–Holland Mathematical Library, vol. 65, Revised edn. North–Holland Publishing Co., Amsterdam (2002)

  2. Dzamonja, M., Shelah, S.: On squares, outside guessing of clubs and \({I_{<f}[\lambda]}\). Fundam. Math. 148, 165–198 (1995). arXiv:math.LO/9510216

  3. Gitik M., Köepke P.: Violating the singular cardinals hypothesis without large cardinals Isr. J. Math. 191, 901–922 (2012)

    MATH  Google Scholar 

  4. Göbel, R., Trlifaj, J.: Approximations and Endomorphism Algebras of Modules, vols. I, II. de Gruyter Expositions in Mathematics. Walter de Gruyter, Berlin/Boston (2012)

  5. Göbel R., Shelah S.: \({\aleph_n}\)-free modules with trivial dual. Results Math. 54, 53–64 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Göbel R., Herden D., Shelah S.: Prescribing endomorphism rings of \({\aleph_n}\)-free modules. J. Eur. Math. Soc. 16, 1775–1816 (2014)

    Article  MATH  Google Scholar 

  7. Göbel R., Shelah S., Struengmann L.: \({\aleph_n}\)-free modules over complete discrete valuation domains with small dual. Glasg. Math. J. 55, 369–380 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Larson P., Shelah S.: Splitting stationary sets from weak forms of choice. Math. Log. Q. 55, 299–306 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Rubin M., Shelah S.: Combinatorial problems on trees: partitions, Δ-systems and large free subtrees. Ann. Pure Appl. Log. 33, 43–81 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  10. Shelah, S.: Proper Forcing, Lecture Notes in Mathematics, vol. 940, Springer, Berlin, xxix+496 pp (1982)

  11. Shelah, S.: Cardinal Arithmetic, Oxford Logic Guides, vol. 29. Oxford University Press, Oxford (1994)

  12. Shelah, S.: Combinatorial Background for Non-structure arXiv:1512.04767 [math.LO]

  13. Shelah, S.: PCF: The Advanced PCF Theorems. arXiv:1512.07063

  14. Shelah, S.: A combinatorial principle and endomorphism rings. I. On p-groups. Isr. J. Math. 49, 239–257 (1984). Proceedings of the 1980/1 Jerusalem Model Theory year

  15. Shelah, S.: Black Boxes. arXiv:0812.0656

  16. Shelah S.: Products of regular cardinals and cardinal invariants of products of Boolean algebras. Isr. J. Math. 70, 129–187 (1990)

    Article  MATH  Google Scholar 

  17. Shelah S.: Reflecting stationary sets and successors of singular cardinals. Arch. Math. Log. 31, 25–53 (1991)

    Article  MATH  Google Scholar 

  18. Shelah, Saharon: Further cardinal arithmetic. Isr. J. Math. 95, 61–114 (1996). arXiv:math.LO/9610226

  19. Shelah, S.: The generalized continuum hypothesis revisited. Isr. J. Math. 116, 285–321 (2000). arXiv:math.LO/9809200

  20. Shelah, S.: Set theory without choice: not everything on cofinality is possible. Arch. Math. Log. 36, 81–125 (1997). A special volume dedicated to Prof. Azriel Levy. arXiv:math.LO/9512227

  21. Shelah, S.: The pcf-theorem revisited. In: Graham, N. (eds) The Mathematics of Paul Erdős, II, Algorithms and Combinatorics, vol. 14, pp. 420–459. Springer (1997). arXiv:math.LO/9502233

  22. Shelah, S.: More on the Revised GCH and the Black Box. Ann. Pure Appl. Log. 140, 133–160 (2006). arXiv:math.LO/0406482

  23. Shelah, S.: PCF without choice. Arch. Math. Log. submitted. arXiv:math.LO/0510229

  24. Shelah, S.: \({\aleph_n}\)-free abelain group with no non-zero homomorphism to \({\mathbb{Z}}\). CUBO Math. J. 9, 59–79 (2007). arXiv:math.LO/0609634

  25. Shelah, S.: Models of expansions of \({\mathbb{N}}\) with no end extensions. Math. Log. Q. 57, 341–365 (2011). arXiv:0808.2960

  26. Shelah, S.: PCF arithmetic without and with choice. Isr. J. Math. 191, 1–40 (2012). arXiv:0905.3021

  27. Shelah S.: Pseudo pcf. Isr. J. Math. 201, 185–231 (2014)

    Article  MATH  Google Scholar 

  28. Shelah, S.: Quite free complicated abelian groups, PCF and Black Boxes, Isr. J. Math. submitted

  29. Shelah, S.: Set theory with weak choice: Ax 4 and Abelian groups (in preparation)

  30. Shelah, S. and Kojman, The PCF trichotomy theorem does not hold for short sequences, Arch. Math. Log. 39, 213–218 (2000)

Download references

Author information

Authors and Affiliations

  1. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, 9190401, Jerusalem, Israel

    Saharon Shelah

  2. Department of Mathematics, The State University of New Jersey, Rutgers, Hill Center - Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Saharon Shelah

Authors
  1. Saharon Shelah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saharon Shelah.

Additional information

Dedicated to the memory of Richard Laver.

References to outside papers like [23, 2.13 = Ls.2] means to 2.13 where s.2 is the label used there, so intended only to help the author if more is added to [23].

The author thanks Alice Leonhardt for the beautiful typing. The author would like to thank the Israel Science Foundation for partial support of this research (Grant No. 1053/11). Publication 1005.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shelah, S. ZF + DC + AX4 . Arch. Math. Logic 55, 239–294 (2016). https://doi.org/10.1007/s00153-015-0469-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-015-0469-0

Keywords

Mathematics Subject Classification

Search

Navigation