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An important question in $\alpha$-recursion theory is whether there is a minimal $\alpha$-degree at $\alpha=\aleph_\omega.$

Question 1. Who first introduced the above question, and where can I find more information about it?

Question 2. Is there any singular cardinal $\alpha$ for which it is known there exists a minimal $\alpha$-degree? Is there any singular cardinal $\alpha$ for which it is known there is no minimal $\alpha$-degree?

Question 3. What can we say about singular cardinals of uncountable cofinality?

Question 4. What can we say about large cardinals (say at least inaccessible)?

Any good references are appreciated.

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    $\begingroup$ Interesting questions! $\endgroup$
    – Mostafa Mirabi
    Jul 25, 2015 at 18:00
  • $\begingroup$ Try looking at some of Sy Friedman early works on admissibility and uncountable admissible ordinals. $\endgroup$
    – William
    Jul 25, 2015 at 19:29
  • $\begingroup$ I think Wei Li (currently at Kurt Goedel Research Centre, logic.univie.ac.at/~liw8) knows about this problem. You may consider asking her about the state of the art and then share it with us. $\endgroup$
    – Dávid Natingga
    Jul 26, 2015 at 20:46

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