4.5 page 16: definition of lambda_2 (R) 4.17 page 23: reduction on a relation of cardinality <= lambda_2 (R) Def 5.1 page 26: defining lambda_3 Definition 5.11A(?) defining K^{word} page 3(?) Fact 5.2 page 26(?) lambda_2 in {lambda_3, lambda_2^+} 5.11,5.12, 5.14 pages 31,32,33: analyses the case lambda_2 not= lambda_3, rest on the case of equality page 35 line-16; defining chi page 35 line-2 }, abstract2 = {6.11 page 40: finishing this case and from now on chi >= cardinality(R) when U has cardinality aleph_0 Explanation of the from of the paper: The original aim of this article was to prove that for every K (a family of relations on U on a fixed arity) its quantifier is equivalent to one for KU, a family of equivalence relations (all such classes are assumed to be closed under isomorphism). Sections 1-6 were written for this and realize it to large extent. Clearly it suffice to deal with I with one isomorphism type as long as the interpretations are uniform. But two essential difficulties arise (1) the quantifier exists^{word}_{lambda} (a well ordering of length lambda), provably is not biinterpretable with exists_K for any family K of equivalence classes. This is not so serious: just add another case. (2) It is consistent that there are cardinals chi, lambda satisfying chi <= lambda <= 2^chi such that R is e.g. a family of chi sunsets of lambda, again we cannot reduce this to equivalence relations in general (see section 8) Only under the assumptions V=L and considering more liberal notion of bi-interpretability (equiv_{exp} rather than equiv_{int} the desired result is gotten. However when the gap degenerates for any reason we get the original hope (note exists^{word}_omega is second order quantification).
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