Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-25T05:32:07.780Z Has data issue: false hasContentIssue false
  • English
  • Français

BPFA and projective well-orderings of the reals

Published online by Cambridge University Press:  12 March 2014

Andrés Eduardo Caicedo  and
Sy-David Friedman
Andrés Eduardo Caicedo
Affiliation:
Boise State University, Department of Mathematics—1910 University Drive, Boise, ID 83725, USA, URL: http://math.boisestate.edu/~caicedo/, E-mail: caicedo@math.boisestate.edu
Sy-David Friedman
Affiliation:
Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Vienna, Austria, URL: http://www.logic.univie.ac.at/~sdf/, E-mail: sdf@logic.univie.ac.at
Get access Rights & Permissions [Opens in a new window]

Abstract

If the bounded proper forcing axiom BPFA holds and ω1 = ω1L, then there is a lightface Σ31 well-ordering of the reals. The argument combines a well-ordering due to Caicedo-Veličković with an absoluteness result for models of MA in the spirit of “David's trick.” We also present a general coding scheme that allows us to show that BPFA is equiconsistent with R being lightface Σ41 for many “consistently locally certified” relations R on ℝ. This is accomplished through a use of David's trick and a coding through the Σ2 stable ordinals of L.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 4 , December 2011 , pp. 1126 - 1136
Copyright
Copyright © Association for Symbolic Logic 2011

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

[1]Bagaria, Joan, Bounded forcing axioms as principles of generic absoluteness. Archive for Mathematical Logic, vol. 39 (2000), no. 6, pp. 393–401.CrossRefGoogle Scholar
[2]Baumgartner, James, Applications of the proper forcing axiom. Handbook of set-theoretic topology (Kunen, Kenneth and Vaughan, Jerry, editors), North-Holland, Amsterdam, 1984, pp. 913–959.CrossRefGoogle Scholar
[3]Caicedo, Andrés E., Projective well-orderings and bounded forcing axioms, this Journal, vol. 70 (2005), no. 2, pp. 557–572.Google Scholar
[4]Caicedo, Andrés E. and Schindler, Ralf, Projective well-orderings of the reals, Archive for Mathematical Logic, vol. 45 (2006), pp. 783–793.CrossRefGoogle Scholar
[5]Caicedo, Andrés E. and Veličković, Boban, The bounded proper forcing axiom and well-orderings of the reals, Mathematical Research Letters, vol. 13 (2006), no. 2–3, pp. 393–408.CrossRefGoogle Scholar
[6]Friedman, Sy-D., David's trick, Sets and proofs (Leeds, 1997) (Cooper, Barry and Truss, John, editors), London Mathematical Society Lecture Note Series, vol. 258, Cambridge University Press, Cambridge, 1999, pp. 67–71.CrossRefGoogle Scholar
[7]Friedman, Sy-D., Fine structure and class forcing, de Gruyter, Series in Logic and its Applications, vol. 3, Walter de Gruyter, Berlin, 2000.CrossRefGoogle Scholar
[8]Friedman, Sy-D. and Schindler, Ralf, Universally Baire sets and definable well-orderings of the reals, this Journal, vol. 68 (2003), no. 4, pp. 1065–1081.Google Scholar
[9]Goldstern, Martin and Shelah, Saharon, The bounded proper forcing axiom, this Journal, vol. 60 (1995), no. 1, pp. 58–73.Google Scholar
[10]Hjorth, Greg, The size of the ordinal u2, Journal of the London Mathematical Society (2), vol. 52 (1995), no. 3, pp. 417–433.CrossRefGoogle Scholar
[11]Martin, Donald A. and Solovay, Robert M., Internal Cohen extensions, Annals of Mathematical Logic, vol. 2 (1970), no. 2, pp. 143–178.CrossRefGoogle Scholar
[12]Moore, Justin, Set mapping reflection. Journal of Mathematical Logic, vol. 5 (2005), no. 1, pp. 87–98.CrossRefGoogle Scholar
[13]Schindler, Ralf, Coding into K by reasonable forcing, Transactions of the American Mathematical Society, vol. 353 (2000), pp. 479–489.CrossRefGoogle Scholar
[14]Schindler, Ralf, Proper forcing and remarkable cardinals II, this Journal, vol. 66 (2001), no. 3, pp. 1481–1492.Google Scholar
[15]Shelah, Saharon, Proper and improper forcing, second ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998.CrossRefGoogle Scholar
[16]Shelah, Saharon and Stanley, Lee J., Coding and reshaping when there are no sharps, Set theory of the continuum (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., vol. 26, Springer, New York, 1992, pp. 407–416.Google Scholar
[17]Zeman, Martin, Inner models and large cardinals, de Gruyter, Series in Logic and its Applications, vol. 5, Walter de Gruyter, Berlin, 2002.CrossRefGoogle Scholar