We generically construct a model in which the ${\mathrm{\Sigma }}_3^1$-separation property is true, i.e. every pair of disjoint ${\mathrm{\Sigma }}_3^1$-sets can be separated by a ${\mathrm{\Delta }}_3^1$-definable set. This answers an old question from the problem list “Surrealist landscape with figures” by A. Mathias from 1968. We also construct a model in which the (lightface) ${\mathrm{\Sigma }}_3^1$-separation property is true.

中文翻译：

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强制 Σ31 分离属性

我们一般构建一个模型，其中${\mathrm{\Sigma }}_3^1$-separation 属性为真，即每一对不相交的${\mathrm{\Sigma }}_3^1$-sets 可以用 a 分隔${\mathrm{\Delta }}_3^1$- 可定义的集合。这回答了 A. Mathias 于 1968 年提出的问题列表“超现实主义风景与人物”中的一个老问题。我们还构建了一个模型，其中 (lightface)${\mathrm{\Sigma }}_3^1$-separation 属性为真。