On a problem of Steve Kalikow

Saharon Shelah

Fundamenta Mathematicae (2000)

  • Volume: 166, Issue: 1-2, page 137-151
  • ISSN: 0016-2736

Abstract

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The Kalikow problem for a pair (λ,κ) of cardinal numbers,λ > κ (in particular κ = 2) is whether we can map the family of ω-sequences from λ to the family of ω-sequences from κ in a very continuous manner. Namely, we demand that for η,ν ∈ ω we have: η, ν are almost equal if and only if their images are. We show consistency of the negative answer, e.g., for ℵ ω but we prove it for smaller cardinals. We indicate a close connection with the free subset property and its variants.

How to cite

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Shelah, Saharon. "On a problem of Steve Kalikow." Fundamenta Mathematicae 166.1-2 (2000): 137-151. <http://eudml.org/doc/212473>.

@article{Shelah2000,
abstract = {The Kalikow problem for a pair (λ,κ) of cardinal numbers,λ > κ (in particular κ = 2) is whether we can map the family of ω-sequences from λ to the family of ω-sequences from κ in a very continuous manner. Namely, we demand that for η,ν ∈ ω we have: η, ν are almost equal if and only if their images are. We show consistency of the negative answer, e.g., for $ℵ_ω$ but we prove it for smaller cardinals. We indicate a close connection with the free subset property and its variants.},
author = {Shelah, Saharon},
journal = {Fundamenta Mathematicae},
keywords = {set theory; forcing; continuity; Kalikow; free subset; Kalikow problem; -sequences; consistency},
language = {eng},
number = {1-2},
pages = {137-151},
title = {On a problem of Steve Kalikow},
url = {http://eudml.org/doc/212473},
volume = {166},
year = {2000},
}

TY - JOUR
AU - Shelah, Saharon
TI - On a problem of Steve Kalikow
JO - Fundamenta Mathematicae
PY - 2000
VL - 166
IS - 1-2
SP - 137
EP - 151
AB - The Kalikow problem for a pair (λ,κ) of cardinal numbers,λ > κ (in particular κ = 2) is whether we can map the family of ω-sequences from λ to the family of ω-sequences from κ in a very continuous manner. Namely, we demand that for η,ν ∈ ω we have: η, ν are almost equal if and only if their images are. We show consistency of the negative answer, e.g., for $ℵ_ω$ but we prove it for smaller cardinals. We indicate a close connection with the free subset property and its variants.
LA - eng
KW - set theory; forcing; continuity; Kalikow; free subset; Kalikow problem; -sequences; consistency
UR - http://eudml.org/doc/212473
ER -

References

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  1. [Ka90] S. Kalikow, Sequences of reals to sequences of zeros and ones, Proc. Amer. Math. Soc. 108 (1990), 833-837. Zbl0692.04007
  2. [Ko84] P. Koepke, The consistency strength of the free-subset property for ω ω , J. Symbolic Logic 49 (1984), 1198-1204. Zbl0592.03042
  3. [Mi91] A. W. Miller, Arnie Miller's problem list, in: H. Judah (ed.), Set Theory of the Reals (Ramat Gan, 1991), Israel Math. Conf. Proc. 6, Bar-Ilan Univ., Ramat Gan, 1993, 645-654. Zbl0828.03017
  4. [Sh 76] S. Shelah, Independence of strong partition relation for small cardinals, and the free-subset problem, J. Symbolic Logic 45 (1980), 505-509. Zbl0453.03052
  5. [Sh 124] S. Shelah, ℵ ω may have a strong partition relation, Israel J. Math. 38 (1981), 283-288. 
  6. [Sh 110] S. Shelah, Better quasi-orders for uncountable cardinals, ibid. 42 (1982), 177-226. Zbl0499.03040
  7. [Sh:b] S. Shelah, Proper Forcing, Lecture Notes in Math. 940, Springer, Berlin, 1982. 
  8. [Sh:g] S. Shelah, Cardinal Arithmetic, Oxford Logic Guides 29, Oxford Univ. Press, 1994. 
  9. [Sh 481] S. Shelah, Was Sierpiński right? III Can continuum-c.c. times c.c.c. be continuum-c.c.? Ann. Pure Appl. Logic 78 (1996), 259-269. Zbl0858.03049
  10. [Sh:F254] S. Shelah, More on Kalikow Property of pairs of cardinals. 
  11. [Sh 513] S. Shelah, PCF and infinite free subsets, Arch. Math. Logic, to appear. 
  12. [Si70] J. Silver, A large cardinal in the constructible universe, Fund. Math. 69 (1970), 93-100. Zbl0208.01503

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