A set $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ is universal for countable subsets of ${\mathbb {R}}$ if and only if for all $x \in {\mathbb {R}}$ , the section $U_x = \{y \in {\mathbb {R}} : U(x,y)\}$ is countable and for all countable sets $A \subseteq {\mathbb {R}}$ , there is an $x \in {\mathbb {R}}$ so that $U_x = A$ . Define the equivalence relation $E_U$ on ${\mathbb {R}}$ by $x_0 \ E_U \ x_1$ if and only if $U_{x_0} = U_{x_1}$ , which is the equivalence of codes for countable sets of reals according to U. The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $E_{U^*}$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $E_U$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $E_U$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $\mathbf {\Sigma }_3^1$ -generic absoluteness, then for all $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ that are $\mathbf {\Sigma }_1^1$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $E_U$ .

集合 $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ 对于 ${\mathbb {R}}$ 的 可数子集是通用的 当且仅当对于所有 $x \in {\ mathbb {R}}$ ，部分 $U_x = \{y \in {\mathbb {R}} : U(x,y)\}$ 是可数的并且对于所有可数集 $A \subseteq {\mathbb {R }}$ ，有一个 $x \in {\mathbb {R}}$ 这样 $U_x = A$ 。通过 $x_0 \ E_U \ x_1$ 定义 ${\mathbb {R}} $上 的等价关系 $E_U$ 当且仅当 $U_{x_0} = U_{x_1}$ ，这是可数集的代码等价根据U的实数 . 等式关系的 Friedman–Stanley 跳跃 $=^+$ 的形式 为 $E_{U^*}$ ，其中 $U^*$ 是最自然的 Borel 集，对于可数集是通用的。主要结果是 $=^+$ 和 $E_U$ 对于任何是 Borel 和可数集通用的U都等价于 Borel 双可约性。对于所有对可数集来说是 Borel 和通用的U， $E_U$ 是 Borel 可双约到 $=^+$ 的 。如果假设 $\mathbf {\Sigma }_3^1$ - 泛型绝对性的特定实例 ，那么对于所有 $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ 这是 $\mathbf {\Sigma }_1^1$ （Borel 集的连续图像）和可数集的通用，有 $=^+$ 的 Borel 减少 到 $E_U$ 。