Cardinality of Equivalence Relations

The number of L∞κ-equivalent non-isomorphic models for κ weakly compact

For a cardinal κ and a model M of cardinality κ let No(M) denote the number of non-isomorphic models of cardinality κ which are L∞,κequivalent to M . We prove that for κ a weakly compact cardinal, the question of the possible values of No(M) for models M of cardinality κ is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are Σ1-definable ...

Cardinality and Equivalence Relations

2 9 N ov 1 99 9 The number of L ∞ κ - equivalent non - isomorphic models for κ weakly compact

For a cardinal κ and a model M of cardinality κ let No(M) denote the number of non-isomorphic models of cardinality κ which are L∞κ-equivalent to M. In [She82] Shelah established that when κ is a weakly compact cardinal and μ ≤ κ is a nonzero cardinal, there exists a model M of cardinality κ with No(M) = μ. We prove here that if κ is a weakly compact cardinal, the question of the possible value...

Generalized voice-leading spaces.

Western musicians traditionally classify pitch sequences by disregarding the effects of five musical transformations: octave shift, permutation, transposition, inversion, and cardinality change. We model this process mathematically, showing that it produces 32 equivalence relations on chords, 243 equivalence relations on chord sequences, and 32 families of geometrical quotient spaces, in which ...

Real Analysis Exchange

For f : [0, 1] → R, consider the relation Ef on [0, 1] defined by (xn)Ef (yn)⇔ P n<ω f(|yn−xn|) <∞. We study the Borel reducibility of Borel equivalence relations of the form Ef . Our results indicate that for every 1 ≤ p < q < ∞, the order ≤B of Borel reducibility on the set of equivalence relations {E : EIdp ≤B E ≤B EIdq} is more complicated than expected, e.g. consistently every linear order...