Summary
The local homogeneity property is defined as in [Mak]. We show thatL ωω(Q1) and some related logics do not have the local homogeneity property, whereas cofinality logicL ωω(Qcfω) has the homogeneity property. Both proofs use forcing and absoluteness arguments.
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Mildenberger, H. On the homogeneity property for certain quantifier logics. Arch Math Logic 31, 445–455 (1992). https://doi.org/10.1007/BF01277486
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DOI: https://doi.org/10.1007/BF01277486