Effectiveness and Vaught’s gap 𝜔 two-cardinal theorem

Schmerl, James H.

doi:10.1090/S0002-9939-1976-0432445-4

Effectiveness and Vaught’s gap $\omega$ two-cardinal theorem
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Proc. Amer. Math. Soc. 58 (1976), 237-240 Request permission

Abstract:

We consider functions $f$ with the property that whenever $\sigma$ is a sentence in ${L_{\omega \omega }}$, then $f(\sigma ) < \omega$, and if $\sigma$ has a gap $> f(\sigma )$ model, then $\sigma$ admits all types. A question of Barwise is answered by showing that no such $f$ is recursive, and that the least such $f$ is not co-r.e.

References

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