A uniqueness theorem for iterations. (English) Zbl 1042.03036
It is a result of Woodin that two different generic ultrapowers of the same model of \(\text{ZFC}+ \text{MA}_{\aleph_1}\) have no new reals in common. The author strengthens this to iterations of arbitrary countable length, in the process providing a shorter proof of Woodin’s original result. The author proves that for \(M\) a countable model of \(\text{ZFC}+ \text{MA}_{\aleph_1}\), for every real \(x\) there is a unique shortest iteration \(j: M\rightarrow N\) with \(x\in N\), or none at all.
Reviewer: J. M. Plotkin (East Lansing)
MSC:
| 03E40 | Other aspects of forcing and Boolean-valued models |
| 03E50 | Continuum hypothesis and Martin’s axiom |
References:
| [1] | The axiom of determinacy, forcing axioms, and the nonstationary ideal 1 (1999) · Zbl 0954.03046 |
| [2] | Mathematical logic and foundations of set theory (Proceedings of the international colloquium, Jerusalem, 1968) pp 84– (1970) |
| [3] | DOI: 10.4064/fm168-1-3 · Zbl 0969.03059 · doi:10.4064/fm168-1-3 |
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