# On a problem of Steve Kalikow

Fundamenta Mathematicae (2000)

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

## Access Full Article

top## Abstract

top## How to cite

topShelah, 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

top- [Ka90] S. Kalikow, Sequences of reals to sequences of zeros and ones, Proc. Amer. Math. Soc. 108 (1990), 833-837. Zbl0692.04007
- [Ko84] P. Koepke, The consistency strength of the free-subset property for ${\omega}_{\omega}$, J. Symbolic Logic 49 (1984), 1198-1204. Zbl0592.03042
- [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
- [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
- [Sh 124] S. Shelah, ${\aleph}_{\omega}$ may have a strong partition relation, Israel J. Math. 38 (1981), 283-288.
- [Sh 110] S. Shelah, Better quasi-orders for uncountable cardinals, ibid. 42 (1982), 177-226. Zbl0499.03040
- [Sh:b] S. Shelah, Proper Forcing, Lecture Notes in Math. 940, Springer, Berlin, 1982.
- [Sh:g] S. Shelah, Cardinal Arithmetic, Oxford Logic Guides 29, Oxford Univ. Press, 1994.
- [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
- [Sh:F254] S. Shelah, More on Kalikow Property of pairs of cardinals.
- [Sh 513] S. Shelah, PCF and infinite free subsets, Arch. Math. Logic, to appear.
- [Si70] J. Silver, A large cardinal in the constructible universe, Fund. Math. 69 (1970), 93-100. Zbl0208.01503

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.