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Shelah's Main Gap Theorem states that for all first-order, complete theories, T, in a countable language, we have that either $$I(T,\aleph_\alpha)=2^{\aleph_\alpha}$$ or $$I(T,\aleph_\alpha)<\beth_{\omega_1}(\alpha)$$ This result, while beautiful, leaves some questions. Can analogues of this result be extended to larger languages? Do other theorems exists for logics like $L_{\omega_1 \omega}$ or $L_{\omega_1 \omega}(Q)$? Do there exist any logics without the 'gap phenomenon'?

Any resources would be helpful. Thanks!

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  • $\begingroup$ It seems that the problem for $L_{\omega_1,\omega}$ is still open, as well as even more modest problems (see the discussion on the end of page 43 here): shelah.logic.at/files/702.pdf $\endgroup$ – Haim Jul 13 '14 at 3:27
  • $\begingroup$ @Haim: Ahh, your right. On page 25, he remarks that it is still a problem to prove the main gap for uncountable theories! $\endgroup$ – Kyle Gannon Jul 13 '14 at 4:04
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Extending Shelah's main gap to non first-order or even to first-order theories in an uncountable language is a major hard open problem.

I am familar with couple of extensions to non f.o. (both papers are available from my web page):

  1. Rami Grossberg and Bradd Hart. The classification theory of excellent classes, Journal of Symbolic Logic, 54, (1989) 1359--1381.
  2. Rami Grossberg and Olivier Lessmann. Abstract decomposition theorem and applications, Contemporary Mathematics, Vol 380, (2005), AMS, pp. 73--108.

For full $L_{\omega_1,\omega}$ this seems to be super hard as it may require proving Shelah's categoricity conjecture for $L_{\omega_1,\omega}$ which despite of many hundreds of pages of approximation is still open.

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