Quite free complicated abelian groups, pcf and black boxes
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QUITE FREE COMPLICATED ABELIAN GROUPS, PCF AND BLACK BOXES
BY
Saharon Shelah∗ Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram The Hebrew University of Jerusalem, Jerusalem, 91904, Israel and Department of Mathematics, Hill Center—Busch Campus Rutgers, The State University of New Jersey 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA e-mail: [email protected] URL: http://shelah.logic.at
ABSTRACT
We would like to build Abelian groups (or R-modules) which on the one hand are quite free, say ℵω+1 -free, and on the other hand are complicated in a suitable sense. We choose as our test problem one having no nontrivial homomorphism to Z (known classically for ℵ1 -free, recently for ℵn -free). We succeed to prove the existence of even ℵω1 ·n -free ones. This requires building n-dimensional black boxes, which are quite free. This combinatorics is of self interest and we believe will be useful also for other purposes. On the other hand, modulo suitable large cardinals, we prove that it is consistent that every ℵω1 ·ω -free Abelian group has non-trivial homomorphisms to Z.
∗ The author thanks the Israel Science Foundation for support of this paper, Grant
No. 1053/11. Publication 1028. The author thanks Alice Leonhardt for the beautiful typing. The reader should note that the version in my website is usually more updated that the one in the mathematical archive. Received December 14, 2014 and in revised form January 24, 2019
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S. SHELAH
Isr. J. Math.
0. Introduction 0(A). Abelian groups. We would like to determine the supremum of all λ for which we can prove TDCλ , so, dually, the minimal λ such that consistently we have NTDCλ which means the failure of TDCλ , the trivial dual conjecture for λ, where: (TDCλ ) there is a λ-free Abelian group G such that Hom(G, Z) = 0. This seems the weakest algebraic statement of this kind; it is consistent that the number is ∞, as if V = L then TDCλ holds for every λ (see, e.g., [GT12]). On the one hand by Magidor–Shelah [MS94], for λ = min{λ : λ is a fixed point, that is λ = ℵλ }, NTDCλ is consistent, as more is proved there: consistently “λ-free ⇒ free”. On the other hand, for a long time we have known the following for λ = ℵ1 , and recently by [She07] we know that for λ = ℵn there are examples using the n-BB (n-dimensional black boxes) introduced there (for every n). Subsequently, those were used for more complicated algebraic relatives in G¨ obel– Shelah [GS09], G¨ obel–Shelah–Str¨ ungman [GSS13] and G¨obel–Herden–Shelah [GHS]. In [She13b] we have several close approximations to proving in ZFC the existence for ℵω , that is TDCℵω using 1-black boxes. Here we finally fully prove that TDCℵω holds and much more; λ = ℵω1 ·ω is the first cardinal for which TDCλ cannot be proved in ZFC. The existence proof for λ < λ is a major result here, relying on the existence proof of quite free nblack boxes (in §1) which use results on pcf (see [She13a]). For complementary consistency results we start with the universe forced in [MS94] and then we force with a c.c.
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