A $\Delta ^2_2$ well-order of the reals and incompactness of $L(Q^{MM})$

by Abraham and Shelah. [AbSh:403]
Annals Pure and Applied Logic, 1993
A forcing poset of size 2^{2^{aleph_1}} which adds no new reals is described and shown to provide a Delta^2_2 definable well-order of the reals (in fact, any given relation of the reals may be so encoded in some generic extension). The encoding of this well-order is obtained by playing with products of Aronszajn trees: Some products are special while other are Suslin trees. The paper also deals with the Magidor-Malitz logic: it is consistent that this logic is highly non compact.

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