Journal of Symbolic Logic

Limit Ultrapowers and Abstract Logics

Paolo Lipparini

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We associate with any abstract logic $L$ a family $\mathbf{F}(L)$ consisting, intuitively, of the limit ultrapowers which are complete extensions in the sense of $L$. For every countably generated $\lbrack\omega, \omega\rbrack$-compact logic $L$, our main applications are: (i) Elementary classes of $L$ can be characterized in terms of $\equiv_L$ only. (ii) If $\mathfrak{U}$ and $\mathfrak{B}$ 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 $\lbrack\omega,\omega\rbrack$-compact and satisfies an interpolation property stronger than unrelativized $\triangle$-closure. (iv) If $L = L_{\omega\omega(Q_\alpha)}$, then $\operatorname{cf}(\omega_\alpha) > \omega$ and $\lambda^\omega < \omega_\alpha$ for all $\lambda < \omega_\alpha$. We also prove that no proper extension of $L_{\omega\omega}$ generated by monadic quantifiers is compact. This strengthens a theorem of Makowsky and Shelah. We solve a problem of Makowsky concerning $L_{\kappa\lambda}$-compact cardinals. We partially solve a problem of Makowsky and Shelah concerning the union of compact logics.

Article information

J. Symbolic Logic, Volume 52, Issue 2 (1987), 437-454.

First available in Project Euclid: 6 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 03C95: Abstract model theory
Secondary: 03C20: Ultraproducts and related constructions 03C80: Logic with extra quantifiers and operators [See also 03B42, 03B44, 03B45, 03B48] 03C52: Properties of classes of models 03C45: Classification theory, stability and related concepts [See also 03C48]


Lipparini, Paolo. Limit Ultrapowers and Abstract Logics. J. Symbolic Logic 52 (1987), no. 2, 437--454.

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