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A version of $$\kappa $$ κ -Miller forcing
Archive For Mathematical Logic ( IF 0.4 ) Pub Date : 2020-02-20 , DOI: 10.1007/s00153-020-00721-y
Heike Mildenberger , Saharon Shelah

We consider a version of \(\kappa \)-Miller forcing on an uncountable cardinal \(\kappa \). We show that under \(2^{<\kappa } = \kappa \) this forcing collapses \(2^\kappa \) to \(\omega \) and adds a \(\kappa \)-Cohen real. The same holds under the weaker assumptions that \({{\,\mathrm{cf}\,}}(\kappa ) > \omega \), \(2^{2^{<\kappa }}= 2^\kappa \), and forcing with \(([\kappa ]^\kappa , \subseteq )\) collapses \(2^\kappa \) to \(\omega \).



中文翻译:

$$ \ kappa $$κ版本-米勒强迫

我们考虑\(\ kappa \)- Miller的版本强制不可数基数\(\ kappa \)。我们显示在\(2 ^ {<\ kappa} = \ kappa \)下,此强制将\(2 ^ \ kappa \)折叠为\(\ omega \)并添加了一个\(\ kappa \)- Cohen real。在\({{\,\ mathrm {cf} \,}}(\ kappa)> \ omega \)\(2 ^ {2 ^ {<\ kappa}} = 2 ^ \ kappa \),并用\(([\ kappa] ^ \ kappa,\ subseteq)\)强制将\(2 ^ \ kappa \)折叠为\(\ omega \)

更新日期:2020-02-20
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