zbMATH — the first resource for mathematics

Forcing with copies of countable ordinals. (English) Zbl 1386.03065
Summary: Let $$\alpha$$ be a countable ordinal and $$\mathbb{P}(\alpha )$$ the collection of its subsets isomorphic to $$\alpha$$. We show that the separative quotient of the poset $$\langle \mathbb{P}(\alpha ), \subset \rangle$$ is isomorphic to a forcing product of iterated reduced products of Boolean algebras of the form $$P(\omega ^\gamma )/\mathcal {I}_{\omega ^\gamma }$$, where $$\gamma \in \mathrm {Lim}\cup \{ 1 \}$$ and $$\mathcal {I}_{\omega ^\gamma }$$ is the corresponding ordinal ideal. Moreover, the poset $$\langle \mathbb{P} (\alpha ), \subset \rangle$$ is forcing equivalent to a two-step iteration of the form $$(P(\omega )/\mathrm {Fin})^+ \ast \pi$$, where $$[\omega ] \Vdash$$ “$$\pi$$ is an $$\omega _1$$-closed separative pre-order” and, if $$\mathfrak{h}=\omega _1$$, to $$(P(\omega )/\mathrm {Fin})^+$$. Also we analyze the quotients over ordinal ideals $$P(\omega ^\delta )/\mathcal {I}_{\omega ^\delta }$$ and the corresponding cardinal invariants $$\mathfrak{h}_{\omega ^\delta }$$ and $$\mathfrak{t}_{\omega ^\delta }$$.

MSC:
 03E40 Other aspects of forcing and Boolean-valued models 03E10 Ordinal and cardinal numbers 03C15 Model theory of denumerable and separable structures 03E35 Consistency and independence results 06A06 Partial orders, general
Full Text:
References:
 [1] [Bla] A. Blass, N. Dobrinen, D. Raghavan, The next best thing to a $$P$$-point, submitted, http://arxiv.org/abs/1308.3790 · Zbl 1367.03078 [2] Fra{\“{\i }}ss{\'”e}, Roland, Theory of relations, Studies in Logic and the Foundations of Mathematics 145, ii+451 pp. (2000), North-Holland Publishing Co.: Amsterdam:North-Holland Publishing Co. · Zbl 0965.03059 [3] Hern{\'a}ndez-Hern{\'a}ndez, Fernando, Distributivity of quotients of countable products of Boolean algebras, Rend. Istit. Mat. Univ. Trieste, 41, 27-33 (2010) (2009) · Zbl 1214.03032 [4] Jech, Thomas, Set theory, Perspectives in Mathematical Logic, xiv+634 pp. (1997), Springer-Verlag: Berlin:Springer-Verlag · Zbl 0882.03045 [5] Kunen, Kenneth, Set theory, Studies in Logic and the Foundations of Mathematics 102, xvi+313 pp. (1980), North-Holland Publishing Co.: Amsterdam:North-Holland Publishing Co. · Zbl 0443.03021 [6] Kurili{\'c}, Milo{\v{s}} S.; Todor{\v{c}}evi{\'c}, Stevo, Forcing by non-scattered sets, Ann. Pure Appl. Logic, 163, 9, 1299-1308 (2012) · Zbl 1250.03102 [7] Kurili{\'c}, Milo{\v{s}} S., From $$A_1$$ to $$D_5$$: towards a forcing-related classification of relational structures, J. Symb. Log., 79, 1, 279-295 (2014) · Zbl 1337.03042 [8] Kurili{\'c}, Milo{\v{s}} S., Posets of copies of countable scattered linear orders, Ann. Pure Appl. Logic, 165, 3, 895-912 (2014) · Zbl 1297.06001 [9] [Kur2] M. S. Kurili\'c, Reduced products, submitted. [10] Laver, Richard, On Fra\“\i ss\'”e’s order type conjecture, Ann. of Math. (2), 93, 89-111 (1971) · Zbl 0208.28905 [11] Rosenstein, Joseph G., Linear orderings, Pure and Applied Mathematics 98, xvii+487 pp. (1982), Academic Press Inc. [Harcourt Brace Jovanovich Publishers]: New York:Academic Press Inc. [Harcourt Brace Jovanovich Publishers] · Zbl 0488.04002 [12] Shelah, Saharon; Spinas, Otmar, The distributivity numbers of finite products of $${\mathcal{P}}(\omega )/\rm fin$$, Fund. Math., 158, 1, 81-93 (1998) · Zbl 0949.03044 [13] [Szym] A. Szyma\'nski, Zhou Hao Xua, The behaviour of $$\omega^2^*$$ under some consequences of Martin’s axiom, General topology and its relations to modern analysis and algebra, V (Prague, 1981), 577-584, Sigma Ser. Pure Math., 3, Heldermann, Berlin, 1983. \endbiblist
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.