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Limit Ultrapowers and Abstract Logics

Paolo Lipparini
The Journal of Symbolic Logic
Vol. 52, No. 2 (Jun., 1987), pp. 437-454
DOI: 10.2307/2274393
Stable URL: http://www.jstor.org/stable/2274393
Page Count: 18
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Limit Ultrapowers and Abstract Logics
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Abstract

We associate with any abstract logic L a family F(L) consisting, intuitively, of the limit ultrapowers which are complete extensions in the sense of L. For every countably generated [ω, ω]-compact logic L, our main applications are: (i) Elementary classes of L can be characterized in terms of $\equiv_L$ only. (ii) If U and B are countable models of a countable superstable theory without the finite cover property, then $\mathfrak{U} \equiv_L \mathfrak{B}$. (iii) There exists the "largest" logic M such that complete extensions in the sense of M and L are the same; moreover M is still [ω,ω]-compact and satisfies an interpolation property stronger than unrelativized ▵-closure. (iv) If L = Lωω(Qα), then $\operatorname{cf}(\omega_\alpha) > \omega$ and $\lambda^\omega < \omega_\alpha$ for all $\lambda < \omega_\alpha$. We also prove that no proper extension of Lωω generated by monadic quantifiers is compact. This strengthens a theorem of Makowsky and Shelah. We solve a problem of Makowsky concerning Lκλ-compact cardinals. We partially solve a problem of Makowsky and Shelah concerning the union of compact logics.

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