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If $r, s\in\mathbb{R}$, we say $r$ is constructible relative to $s$ - and write $r\le_cs$ - if $r\in L[s]$. Modding out by the induced equivalence relation $\equiv_c$, we get a partial order, the degrees of constructibility $\mathcal{C}$. This poset is extremely dependent on the ambient set theory. If $V=L$, the poset is trivial, whereas the existence of a Cohen real automatically leads to a very complicated structure (see, e.g., the paper "The degrees of constructibility of Cohen reals" by Abraham and Shore).

My question is basically, what is known about the possible posets which $\mathcal{C}$ could be? I am especially interested in properties of $\mathcal{C}$ which follow from large cardinals. For example, if $0^\sharp$ exists, then we can deduce that $\mathcal{C}$ has size $2^{\aleph_0}$, which is the maximum possible; presumably this actually gives us a lot more.

As a sub-question, what is a good source for learning about techniques for building a model of $ZFC$ in which $\mathcal{C}$ has some prescribed properties?

ADDED: I'm most interested in what can be said when the size of $\mathcal{C}$ is assumed to be $2^{\aleph_0}$. As a particular example, the paper "Hinges and automorphisms of the degrees of non-constructibility" by P. Farrington (http://citeseerx.ist.psu.edu/viewdoc/summary?doi= has, as Lemma 2.5, the following (CAVEAT: Farrington's paper actually gives the wrong definition of "hinge," which was corrected in Lubarsky's review in the JSL in 1989):

Suppose that $\omega_2^L<\omega_1$. Then there is no nontrivial automorphism of $\mathcal{C}$.

(Now $\omega_2^L<\omega_1$ (and much more!) is a consequence of the existence of $0^\sharp$, so this is a very weak hypothesis.) Results like this are exactly what I'm looking for: properties $\mathcal{C}$ which follow from large cardinals, and in particular can hold under the assumption $\vert\mathcal{C}\vert=2^{\aleph_0}$.

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It's a bit off topic, but does this excitement to see the names of your teachers quoted by others ever go away? :-) –  Asaf Karagila Feb 22 '13 at 1:12
Not even a bit. :-) –  Noah S Feb 22 '13 at 2:22

2 Answers 2

In addition to the work on what can occur in the constructibility degrees, there's a theorem of Bob Lubarsky about what cannot occur. If the lattice of constructibility degrees is countable and has a top element, then it must be a complete lattice. [Lattices of c-degrees, Ann. Pure Appl. Logic 36 (1987) 115-118]

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That's neat! Is there a good intuitive reason why that's the case? –  Noah S Feb 22 '13 at 2:23
ALthough the paper is quite short, it looks to me like a clever and rather technical construction (or rather combination of more than one construction). I don't see a really intuitive way to view it. –  Andreas Blass Feb 22 '13 at 16:22

The article Initial segments of the degrees of constructibility by Marcia Groszek and Richard Shore (Israel Journal of Mathematics June 1988, Volume 63, Issue 2, pp 149-177) shows that

Any constructible, constructibly countable, (dual) algebraic lattice is isomorphic to the degrees of constructibility of reals in some generic extension of L.

And there is a lot of further work on this topic by Groszek, Slaman and others.

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