Usual math sets have special types: countable, compact, open, occasionally Borel, rarely projective, etc. Each such set is described by a single Set Theory formula with parameters unrelated to other formulas. Exotic expressions involving sets related to formulas of unlimited quantifiers height appear mostly in esoteric or foundational studies. Recognizing internal to math (formula-specified) and external (based on parameters in those formulas) aspects of math objects greatly simplifies foundations. I postulate external sets (not internally specified, constituting the domain of variables) to be hereditarily countable and independent of formula-defined classes, i.e. with finite algorithmic information about them. This allows to eliminate all non-integer quantifiers in Set Theory sentences. All with seemingly no need to change almost anything in mathematical papers, only to reinterpret some formalities.

翻译：通常数学集合具有特殊类型：可数集、紧致集、开集、偶尔出现的博雷尔集、罕见的射影集等。每个此类集合由单个集合论公式描述，其参数与其他公式无关。涉及带无界量词高度公式的集合的奇异表达式大多出现在深奥或基础研究中。识别数学对象的内部（公式指定的）与外部（基于这些公式中的参数）方面可极大简化基础。我假定外部集合（非内部指定，构成变量论域）是遗传可数的，且不依赖于公式定义的类，即具备关于它们的有限算法信息。这使得集合论语句中可以消除所有非整数量词。所有这些似乎无需改变数学论文中的几乎任何内容，只需重新解释某些形式化约定。