New reals: Can live with them, can live without them

Judah and Shelah [3] proved that countable support iterations of proper forcings preserve the ω-bounding property (see 2.2 here). In his book Proper and Improper Forcing [8, XVIII §3] Shelah gave several cases of general preservation theorems for proper countable support iterations (the proofs tend to be hard to digest, though). In this paper we deal with “Case A”. A simplified version of this case appeared in Section 5 of the first author’s Tools for your forcing constructions [2]. This version uses the additional requirement that every iterand adds a new real. Note that this requirement is met in most applications, but the case of forcings “not adding reals” has important applications as well (and note that not adding reals is generally not preserved under proper countable support iterations). A proof of the iteration theorem without this additional requirement appeared in [5] and was copied into Set Theory of the Reals [1] (as “first preservation theorem” 6.1.B), but Schlindwein pointed out a problem in this proof. In this paper, we generalize the proof of [2]. We thank Chaz Schlindwein for finding the problems in the existing proofs and bringing them to our attention.