We reformulate slightly Russell’s notion of typicality, so as to eliminate its circularity and make it applicable to elements of any first-order structure. We argue that the notion parallels Martin-Löf (ML) randomness, in the sense that it uses definable sets in place of computable ones and sets of “small” cardinality (i.e., strictly smaller than that of the structure domain) in place of measure zero sets. It is shown that if the domain M satisfies cf(|M |) > א0, then there exist |M | typical elements and only < |M | non-typical ones. In particular this is true for the standard model R of second-order arithmetic. By allowing parameters in the defining formulas, we are led to relative typicality, which satisfies most of van Lambalgen’s axioms for relative randomness. However van Lambalgen’s theorem is false for relative typicality. The class of typical reals is incomparable (with respect to ⊆) with the classes of ML-random, Schnorr random and computably random reals. Also the class of typical reals is closed under Turing degrees and under the jump operation (both ways). Mathematics Subject Classification (2010): 03C98, 03D78

中文翻译：

---

罗素作为另一个随机性概念的典型性

我们稍微重新表述了罗素的典型性概念，以消除其循环性并使其适用于任何一阶结构的元素。我们认为该概念与 Martin-Löf (ML) 随机性相似，因为它使用可定义的集合代替可计算的集合和“小”基数的集合（即，严格小于结构域的基数）代替度量零集。证明如果域M满足cf(|M |) > א0，则存在|M | 典型元素且仅 < |M | 非典型的。对于二阶算术的标准模型 R 尤其如此。通过在定义公式中允许参数，我们得到了相对典型性，这满足了范兰巴尔根关于相对随机性的大部分公理。然而，van Lambalgen 的定理对于相对典型性来说是错误的。典型实数类与 ML 随机、Schnorr 随机数和可计算随机实数类是不可比的（就 ⊆ 而言）。此外，典型实数类在图灵度和跳跃操作（两种方式）下都是封闭的。数学学科分类（2010）：03C98、03D78