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Unbounded and dominating reals in Hechler extensions. (English) Zbl 1278.03083
Standard Hechler \({\mathbb{D}}\) forcing consists of conditions \((s,f)\) with \(s \in \omega^{<\omega}\), \(f \in \omega^\omega\) and \(s \subset f\). \((s',f')\) is stronger than \((s,f)\) if \(s' \supseteq s\) and \(f'\) dominates \(f\) everywhere. \({\mathbb{D}}\) is a \(\sigma\)-centered forcing notion which generically adds a dominating real. A very similar forcing notion, tree Hechler forcing \({\mathbb{D}}_{\mathrm{tree}}\), has as conditions subtrees \(T\) of \(\omega^{<\omega}\) with a stem \(s\) such that for all \(t \in T\) extending \(s\), all but finitely many successors \(t \frown n\) also belong to \(T\). \(T'\) is stronger than \(T\) if \(T' \subseteq T\). \({\mathbb{D}}_{\mathrm{tree}}\) also is \(\sigma\)-centered and adds a dominating real. In fact, \({\mathbb{D}}_{\mathrm{tree}}\) completely embeds into (= is a subforcing of) \({\mathbb{D}}\) and vice versa.
The author shows that the two forcing notions are not equivalent, thus exhibiting an instance of the failure of the natural Cantor-Bernstein theorem for equivalence of forcing notions. More explicitly, in the tree Hechler extension, every unbounded real is infinitely often above some dominating real, while in the classical Hechler extension, there is an unbounded real eventually dominated by every dominating real. To obtain the latter result, the author proves that if \(d\) is the \({\mathbb{D}}\)-generic real, then for every dominating real \(y\) in \(V[d]\) there are non-decreasing functions \(z_0\) and \(z_1\) in the ground model such that \(y\) eventually dominates \(z_0 \circ d \circ z_1\). If, given a forcing extension \(V[G]\), we write \(f \preceq g\) iff there are non-decreasing \(z_0 , z_1 \in V\) such that \(z_0 \circ f \circ z_1 \leq^* g\), this means that \(d\) is a \(\preceq\)-least dominating real in \(V[d]\). The author also discusses ccc extensions adding dominating reals which contain no \(\preceq\)-least reals.
As an application of his results, the author answers a question of C. Laflamme [Math. Log. Q. 40, No. 2, 207–223 (1994; Zbl 0835.03015)] on the structure of the set of functions dominating a given bounded family. He also negatively settles a conjecture of B. Löwe and the reviewer [J. Math. Soc. Japan 63, No. 1, 137–151 (2011; Zbl 1215.03059)] about the structure of intermediate extensions of the Hechler extension.

MSC:
03E40 Other aspects of forcing and Boolean-valued models
03E17 Cardinal characteristics of the continuum
03E35 Consistency and independence results
03E50 Continuum hypothesis and Martin’s axiom
03E55 Large cardinals
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