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In this paper we consider whether L([w) has "enough information" to contain a counterexample to the continuum hypothesis. We believe this question provides deep insight into the difficulties surrounding the continuum hypothesis. We show sufficient conditions for L(W) not to contain such a counterexample. Along the way we establish many results about nonstationary towers, non-reflecting stationary sets, generalizations of proper and semiproper forcing and Chang's conjecture.doi:10.1016/0168-0072(94)00031-w fatcat:4wtoduxe3vaw5cw6iz3h4tpjz4