Killing Luzin and Sierpinski sets. (English) Zbl 0797.03050

Judah, H.; Shelah, Saharon

Proc. Am. Math. Soc. 120, No. 3, 917-920 (1994).

A set of reals \(X\) is called a Luzin set iff it is uncountable and meets every meager set in a countable set. Similarly a set of reals \(X\) is called a Sierpiński set iff it is uncountable and meets every Lebesgue measure zero set in a countable set. Assuming the continuum hypothesis both kinds of sets exist. Clearly a Luzin set cannot be meager and a Sierpiński set cannot have measure zero. Thus if these sets exist there must be nonmeager and nonmeasure zero sets of reals of cardinality \(\omega_ 1\). In this paper a question of Steprāns is answered by showing that it is relatively consistent with ZFC that there are no Luzin or Sierpiński sets but there is a set of reals of cardinality \(\omega_ 1\) which is nonmeager and not of measure zero. The model employed is a countable support iteration of superperfect tree forcing.

Reviewer: A.W.Miller (Madison)

MSC:

03E35	Consistency and independence results
03E15	Descriptive set theory

Keywords:

Luzin set; meager set; Sierpiński set; measure zero set; countable support iteration of superperfect tree forcing