Abstract
It is common knowledge in the set theory community that there exists a duality relating the commutative \(C^*\)-algebras with the family of \(\mathsf {B}\)-names for complex numbers in a boolean valued model for set theory \(V^{\mathsf {B}}\). Several aspects of this correlation have been considered in works of the late 1970s and early 1980s, for example by Takeuti (Two Applications of Logic to Mathematics. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, Kanô Memorial Lectures, vol 3. Publications of the Mathematical Society of Japan, No. 13, 1978) and Fourman et al. (eds.) (Applications of sheaves. In: Lecture Notes in Mathematics, vol 753. Springer, Berlin, 1979), and by Jech (Trans Am Math Soc 289(1):133–162, 1985). Generalizing Jech’s results, we extend this duality so as to be able to describe the family of boolean names for elements of any given Polish space Y (such as the complex numbers) in a boolean valued model for set theory \(V^\mathsf {B}\) as a space \(C^+(X,Y)\) consisting of functions f whose domain X is the Stone space of \(\mathsf {B}\), and whose range is contained in Y modulo a meager set. We also outline how this duality can be combined with generic absoluteness results in order to analyze, by means of forcing arguments, the theory of \(C^+(X,Y)\).
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Notes
X is extremally disconnected if the closure of an open set is open, or equivalently if its regular open sets are closed. Y is Polish if it is a separable topological space whose topology can be induced by a complete metric on Y.
Recall that \(A\subseteq X\) is meager if it is the union of countably many nowhere dense sets, and B is comeager in U if \(U{\setminus } B\) is meager. It requires an argument based on the fact that R is Borel to show that \(R^X/p\) is well defined.
See [10, 19] for the main results of Woodin regarding generic invariance of second order number theory, the second author’s papers [1, 15, 17] for the extension of these results to large fragments of third order number theory, and [2, 3, 20] for a survey of results on generic absoluteness at the highest levels of the set theoretic hierarchy.
Recall that \(H_{\omega _1}\) is the family of hereditarily countable sets. For what concerns us, the relevant observation is that any Polish space is a definable class (with parameters) in \(H_{\omega _1}\).
\(\dot{U_n}\) denotes the \(\mathsf {B}\)-name for the complex numbers in the open ball of the generic extension determined by the rational coordinates and rational radius of the ball \(U_n\).
If \(U_n=Y\cap B_r(q)\) and G is V-generic for \(\mathsf {B}\), \(\dot{U_n}\) denotes the \(\mathsf {B}\)-name for the elements in the Hilbert cube of V[G] belonging to
$$\begin{aligned} \bigcap _{n\in \mathbb {N}}\bigcup \{(B_{r_{mn}}(q_{mn}))^{V[G]}:m\in \mathbb {N}\} \cap (B_r(q))^{V[G]}, \end{aligned}$$where \((B_{r}(q))^{V[G]}\) is the ball in the Hilbert cube \(\mathcal {H}^{V[G]}\) of rational radius r and center q as computed in V[G].
This isomorphism of \(\mathbb {C}^\mathsf {B}\) and \(C^+(St(\mathsf {B}))\) has also been independently proven by Ozawa in [11], but Jech’s proof is in our eyes more elegant and informative.
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Acknowledgements
The second author acknowledges support from the PRIN2012 Grant “Logic, Models and Sets” (2012LZEBFL), the GNSAGA, and the Junior PI San Paolo Grant 2012 NPOI (TO-Call1-2012-0076). This research was completed whilst the second author was a visiting fellow at the Isaac Newton Institute for Mathematical Sciences in the programme “Mathematical, Foundational and Computational Aspects of the Higher Infinite” (HIF) funded by EPSRC Grant EP/K032208/1. We thank the referee for the careful revision of the original submission.
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Vaccaro, A., Viale, M. Generic absoluteness and boolean names for elements of a Polish space. Boll Unione Mat Ital 10, 293–319 (2017). https://doi.org/10.1007/s40574-017-0124-2
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DOI: https://doi.org/10.1007/s40574-017-0124-2