直观地说，
Projective sets, intuitionistically
摘要：
研究Baire空间$\mathcal{N}$的可定义子集.我们论证的逻辑是直观的，我们使用L. E. J.~ Brouwer关于$\mathcal{N}$中的小节的论文及其连续性公理。我们避免了取$\mathcal{N}$子集的补数的操作。$\mathcal{N}$的子集是$\mathbfΣ^1_1$或：解析的，如果它是$\mathcal{N}$的闭子集的投影。重要的$\mathbfΣ^1_1$集是正不可数的$\mathcal{N}$的所有封闭和定位子集的代码集，以及包含至少一个编码$\mathbb{N}$的（正）无限子集的成员的$\mathcal{N}$的所有定位和封闭子集的代码集。$\mathcal{N}$的子集是严格解析的，如果它是$\mathcal{N}$的一个闭的和定位的子集的投影。Brouwer的论文酒吧在$\mathcal{N}$证明分离和有界定理严格解析子集的$\mathcal{N}$。$\mathcal{N}$的子集是$\mathbfΠ^1_1$或：如果它是$\mathcal{N} \times \mathcal{N}=\mathcal{N}$的开子集的共投影，则它是共分析的。在解析集合和共解析集合之间不存在像经典描述集合论那样的对称性。一个重要的$\mathbfΠ^1_1$集是$\mathcal{N}$的所有闭子集和定位子集的码的集合，其所有成员编码$\mathbb{N}$的几乎有限子集。$\mathcal{N}$的闭子集和定位子集的代码的集合是几乎可数的，或者等价地，在康托意义下是可约的，在一定的长度上被处理。此设置可能不是$\mathbfΠ^1_1$。投影层次崩溃：每个（正）投影集是$\mathbfΣ^1_2$：$\mathcal{N}$的共解析子集的投影。
英文摘要：
We study `definable' subsets of Baire space $\mathcal{N}$. The logic of our arguments is intuitionistic and we use L.E.J.~Brouwer's Thesis on bars in $\mathcal{N}$ and his continuity axioms. We avoid the operation of taking the complement of a subset of $\mathcal{N}$. A subset of $\mathcal{N}$ is $\mathbfΣ^1_1$ or: analytic if it is the projection of a closed subset of $\mathcal{N}$. Important $\mathbfΣ^1_1$ set are the set of the codes of all closed and located subsets of $\mathcal{N}$ that are positively uncountable and the set of the codes of all located and closed subsets of $\mathcal{N}$ containing at least one member coding a (positively) infinite subset of $\mathbb{N}$. A subset of $\mathcal{N}$ is strictly analytic if it is the projection of a closed and located subset of $\mathcal{N}$. Brouwer's Thesis on bars in $\mathcal{N}$ proves separation and boundedness theorems for strictly analytic subsets of $\mathcal{N}$. A subset of $\mathcal{N}$ is $\mathbfΠ^1_1$ or: co-analytic if it is the co-projection of an open subset of $\mathcal{N} \times \mathcal{N}=\mathcal{N}$. There is no symmetry between analytic and co-analytic sets like in classical descriptive set theory. An important $\mathbfΠ^1_1$ set is the set of the codes of all closed and located subsets of $\mathcal{N}$ all of whose members code an almost-finite subset of $\mathbb{N}$. The set of the codes of closed and located subsets of $\mathcal{N}$ that are almost-countable, or, equivalently, \{reducible in Cantor's sense, is treated at some length. This set is probably not $\mathbfΠ^1_1$. The projective hierarchy collapses: every (positively) projective set is $\mathbfΣ^1_2$: the projection of a co-analytic subset of $\mathcal{N}$.
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