Iteration trees.

*(English)*Zbl 0808.03035This paper gives a further report on work done some years ago, which has led to relatively complete understanding of the relationship between the consistency of determinacy and large cardinals. The main theorem announced is: Suppose there are \(n\) Woodin cardinals, \(n<\omega\). Then there is a model \(M_ n\) which is a model of ZFC plus “There are \(n\) Woodin cardinals” plus “\(R\) has a \(\Delta^ 1_{n+2}\) well-order”. This is complementary to the same authors’ paper [ibid. 2, 71-125 (1989; Zbl 0668.03021)], where they showed that if there are \(n\) Woodin cardinals with a measurable cardinal above, then \(\Pi^ 1_{n+1}\) determinacy holds; together these results show that the large cardinal hypotheses used cannot be substantially improved.

The iteration trees of the title are an important ingredient in the proof. This has to do with extending the work of Mitchell and others [see e.g. W. J. Mitchell, Math. Proc. Camb. Philos. Soc. 95, 229-260 (1984; Zbl 0539.03030)], to Woodin cardinals (which are defined in the paper). A basic problem in this work is the comparison of pre-mice; certain of the pre-mice will be mice which together build a core model. In the earlier work, the comparison process resulted in a linear order, or linear iteration of the process of building ultrapowers. A major advance, which is the main report of this paper, is the introduction of tree orderings for the comparison process and hence the tree iterations. Just the case \(n=1\) of the main theorem is proved here.

There is a very readable six page introduction giving a good overview of the work, much of which dates to 1986, and of further advances and problems.

The iteration trees of the title are an important ingredient in the proof. This has to do with extending the work of Mitchell and others [see e.g. W. J. Mitchell, Math. Proc. Camb. Philos. Soc. 95, 229-260 (1984; Zbl 0539.03030)], to Woodin cardinals (which are defined in the paper). A basic problem in this work is the comparison of pre-mice; certain of the pre-mice will be mice which together build a core model. In the earlier work, the comparison process resulted in a linear order, or linear iteration of the process of building ultrapowers. A major advance, which is the main report of this paper, is the introduction of tree orderings for the comparison process and hence the tree iterations. Just the case \(n=1\) of the main theorem is proved here.

There is a very readable six page introduction giving a good overview of the work, much of which dates to 1986, and of further advances and problems.

Reviewer: F.R.Drake (Leeds)

##### Keywords:

consistency of determinacy; large cardinals; Woodin cardinals; iteration trees; comparison of pre-mice; tree orderings
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\textit{D. A. Martin} and \textit{J. R. Steel}, J. Am. Math. Soc. 7, No. 1, 1--73 (1994; Zbl 0808.03035)

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##### References:

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