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：通常的数学集合具有特殊类型：可数、紧致、开集，偶尔为博雷尔集，极少情况下为射影集等。此类集合中的每一个都可由一个集合论公式描述，且该公式的参数与其他公式无关。涉及与无限量词高度公式相关的集合的奇异表达式，主要出现在深奥或基础性研究中。认识到数学对象内部（由公式指定）与外部（基于这些公式中的参数）的区分，可极大简化数学基础。我假设外部集合（非内部指定，构成变量的定义域）是遗传可数的，且独立于公式定义的类，即关于它们仅具有有限的算法信息。这允许消除集合论语句中所有非整数量词。所有这些似乎几乎无需改变数学论文中的任何内容，仅需重新解释某些形式化表述。