# Hechler’s theorem for the null ideal

@article{Burke2004HechlersTF,
title={Hechler’s theorem for the null ideal},
author={Maxim R. Burke and Masaru Kada},
journal={Archive for Mathematical Logic},
year={2004},
volume={43},
pages={703-722}
}
• Published 15 November 2002
• Mathematics
• Archive for Mathematical Logic
Abstract.We prove the following theorem: For a partially ordered set Q such that every countable subset of Q has a strict upper bound, there is a forcing notion satisfying the countable chain condition such that, in the forcing extension, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler’s classical result in the theory of forcing. The corresponding theorem for the meager ideal was established by…
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## References

SHOWING 1-10 OF 20 REFERENCES
A proof of Hechler's theorem on embedding $\aleph_1$-directed sets cofinally into $(\omega^\omega,<^*)$
A proof of Hechler's theorem that any $\aleph_1$-directed partial order can be embedded via a ccc forcing notion cofinally into $\omega^\omega$ ordered by eventual dominance is given.
Set Theory: On the Structure of the Real Line
• Education
• 1995
This research level monograph reflects the current state of the field and provides a reference for graduate students entering the field as well as for established researchers.
On the existence of certain cofinal subsets of ω
• T. Jech, editor, Axiomatic Set Theory, Proc. Symp. Pure Math., pages 155–173. Amer. Math. Soc.
• 1974
A proof of Hechler’s theorem on embedding א1-directed sets cofinally into (ω, <)
• Arch. Math. Logic,
• 1997
Invariants of measure and category. Handbook of Set Theory (in preparation)
• Invariants of measure and category. Handbook of Set Theory (in preparation)
A proof of Hechler’s theorem on embedding א1-directed sets cofinally into (ω
• <). Arch. Math. Logic, 36:399–403
• 1997
Kitami Institute of Technology 165 Koen-cho, Kitami, Hokkaido 090-8507 JAPAN E-mail
• Kitami Institute of Technology 165 Koen-cho, Kitami, Hokkaido 090-8507 JAPAN E-mail
Mad families and iteration theory
• Logic and algebra, 1–31, Contemp. Math., 302, Amer. Math. Soc., Providence, RI
• 2002