Generic absoluteness and the continuum
Generic absoluteness for a class of forcings states that \((H(\aleph_2),\in)\prec_1 (V^B,\in)\) whenever \(V^B\) is a Boolean-valued extension of the universe by a complete Boolean algebra from that class. The main result of the paper is that generic absoluteness for Boolean-valued extensions that preserve stationary subsets of \(\omega_1\) implies the continuum has a well-order of length \(\omega_2\) (definable in \(H(\aleph_2)\) from a single \(\omega_1\)-sequence of reals). Given the one-to-one sequence \(\langle r_\alpha:\alpha\in\omega_1\rangle\) of elements of the Cantor set the coding of subsets of \(\omega\) by ordinals in \(\omega_2\) proceeds as follows. To every countable set of ordinals \(X\) one associates the real \(r_X=r_{\text{otp} X}\). Furthermore the first-difference function \(\Delta(x,y)=\min\{n:x(n)\neq y(n)\}\) on the Cantor set is employed to associate (potentially) to every subset \(S\) of \(\omega_1\) three ordinals \(\gamma>\beta>\alpha\geq\omega_1\) where \(\gamma\) is written as the increasing union of a continuous and increasing sequence \(\langle N_\nu:\nu\in\omega_1\rangle\) of countable sets, all such that \(N_\nu\cap\omega_1\in S\) if and only if \(\Delta(r_{N_\nu\cap\alpha},r_{N_\nu\cap\beta})\) is the largest of the values \(\Delta(u,v)\), where \(u\) and \(v\) can be \(r_{N_\nu\cap\alpha}\), \(r_{N_\nu\cap\beta}\) and \(r_{N_\nu}\), respectively. Now if \((\gamma,\beta,\alpha)\) were associated to two sets \(S\) and \(T\) in this way, with sequences \(\langle N_\nu\rangle_\nu\) and \(\langle M_\nu\rangle_\nu\) respectively, then, since \(N_\nu=M_\nu\) for a cub set of \(\nu\), one would find that \(S\) and \(T\) were equal modulo the cub filter. As one can find \(2^{\aleph_1}\) inequivalent sets, the possibility of assigning a triple and sequence to every subset of \(\omega_1\), abbreviated \(\theta_{\text{AC}}\) in the paper, would imply \(2^{\aleph_1}=\aleph_2\). It turns that one can find for each \(S\) a generic extension in which stationary subsets of \(\omega_1\) are preserved and in which a triple and sequence can be found -- generic absoluteness yields the existence of these objects in the real world and thus the equality \(2^{\aleph_1}=\aleph_2\). Furthermore, the principle \(\theta_{\text{AC}}\) violates the weak diamond of \textit{K. J. Devlin} and \textit{S. Shelah} [Isr. J. Math. 29, 239-247 (1978; Zbl 0403.03040)], hence it implies \(2^{\aleph_0}=2^{\aleph_1}\). A careful inspection of the proof will show how to define a well-order of \(\mathbb R\) in \(H(\aleph_2)\). It seems unknown whether the parameter \(\langle r_\alpha:\alpha\in\omega_1\rangle\) can be replaced by something more effective, e.g., a countable sequence of ordinals -- as one can construct a nonmeasurable set of reals from an \(\omega_1\)-sequence of reals [\textit{S. Shelah}, Isr. J. Math. 48, 1-47 (1984; Zbl 0596.03055)], one could go so far as to argue that the ``more'' above is a bit misplaced.
scientific article; zbMATH DE number 1886041