Evasion and prediction — — the Specker phenomenon and Gross spaces
نویسنده
چکیده
We study the set–theoretic combinatorics underlying the following two algebraic phenomena. (1) A subgroup G ≤ Z exhibits the Specker phenomenon iff every homomorphism G → Z maps almost all unit vectors to 0. Let se be the size of the smallest G ≤ Z exhibiting the Specker phenomenon. (2) Given an uncountably dimensional vector space E equipped with a symmetric bilinear form Φ over an at most countable field K, (E,Φ) is strongly Gross iff for all countably– dimensional U ≤ E, we have dim(U) ≤ ω. Blass showed that the Specker phenomenon is closely related to a combinatorial phenomenon he called evading and predicting. We prove several additional results (both theorems of ZFC and independence proofs) about evading and predicting as well as se, and relate a Luzin–style property associated with evading to the existence of strong Gross spaces. ∗ The author would like to thank the MINERVA-foundation for supporting him
منابع مشابه
Evasion and prediction II
A subgroup G ≤ Z exhibits the Specker phenomenon if every homomorphism G → Z maps almost all unit vectors to 0. We give several combinatorial characterizations of the cardinal se, the size of the smallest G ≤ Z exhibiting the Specker phenomenon. We also prove the consistency of b < e, where b is the unbounding number and e the evasion number. Our results answer several questions addressed by Bl...
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