Hechler's theorem for the meager ideal

Abstract

We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the meager 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.

Around cofin

“…Proof. (2) In both cases the proof is essentially the same, so let us argue for the Solovay algebra only. Represent κ as the disjoint union κ…”

“…If follows from results of Bartoszyński and Kada [2] (for the meager ideal) and Burke and Kada [4] (for the null ideal) that for any cardinals κ and λ of uncountable cofinality we may force that M has a κ × λ-basis, and we may also force that N has a κ × λ-basis. In [6, Theorem 3.7] we constructed a model in which both ideals have κ × λ-bases.…”

“…⊠) 4 Q ζ+1 decides the value of T ˜(ζ, c ˜ζ) and forces (in Y λ κ ) that it is a P ′ α(ζ+1)name. The construction is clearly possible by Lemma 3.12(2,4). Then lettingα * = sup(α(ζ) : ζ < µ) we have that Q µ = P β , Q ˜′ β : β < α * ∈ Y λ κ is a condition stronger than all Q ζ (for ζ < µ); remember 3.12(4) again.…”

Hechler's Theorem for tall analytic P-ideals

“…Theorem 1.3. ( [1]) Let (Q, ≤) be a σ-directed partially ordered set. Then there is a ccc forcing notion P such that in V P a conal subset of (M, ⊆) is order isomorphic to (Q, ≤).…”

Hechler’s theorem for the null ideal

“…In this context, Soukup asked if the statement of Hechler's theorem holds for the meager ideal or the null ideal of the real line with respect to set-inclusion. Bartoszyński and Kada [3] answered the question positively for the meager ideal. In the present paper, we will give a positive answer for the null ideal.…”