AI 中文总结
该研究固定系统函数ϑ构造实数的函数模型，证明其为戴德金完备有序域，不同ϑ对应不可区分的实数集，且可通过命名适配问题，推导得出特定进制下有理数的判定条件及康托尔无理数准则的新来源。
AI 中文摘要
一个实数可以有多个名称：0.5和0.4999…表示同一个数，哪些数具有这种加倍（即多个名称）取决于所用的进制。我们将这种依赖性作为研究对象。固定一个“系统函数”ϑ，它为每个位置独立分配一个进制ϑ(n)≥2，我们将实数构造为数字函数f:ω→ω的等价类，等价关系由两种局部进位操作“收缩”和“拓宽”生成，并通过有限初始段上的一致性来检验。每个等价类最多包含两个规范表示（符号表示除外），即其“主”和“次”辅助函数，因此上述加倍是该理论的定理，而非强加给它的约定。我们定义序、加法和乘法，并验证戴德金完备有序域的公理，因此根据范畴性，ϑ的每一种选择都给出实数集ℝ本身。这些模型作为有序域是不可区分的，仅在元素的命名方式上存在差异，且由于ϑ遍历不可数参数空间，这种命名可被调整以适配特定问题。我们展示了这带来的好处：在其部分乘积吸收所有分母的任意进制中，非零实数是有理数当且仅当它有第二个名称，且康托尔1869年针对康托尔级数的无理数准则可从表示理论而非数论中推导得出。
英文摘要
A real number can have more than one name: $0.5$ and $0.4999\ldots$ denote the same thing, and which numbers enjoy such a doubling depends on the base one writes in. We make that dependence the object of study. Fixing a system function $\vartheta$, which assigns a base $\vartheta(n)\geq2$ to every position independently, we build the real numbers as equivalence classes of digit functions $f:ω\rightarrowω$, where the equivalence is generated by two local carrying moves, contraction and broadening, and tested by agreement on finite initial segments. Each class turns out to contain at most two canonical representatives (up to a sign representation), its primary and secondary auxiliary functions, so the doubling above is a theorem of the theory rather than a convention imposed on it. We define order, addition and multiplication and verify the axioms of a Dedekind-complete ordered field, so that by categoricity every choice of $\vartheta$ delivers $\mathbb{R}$ itself. The models are therefore indistinguishable as ordered fields and differ only in how their elements are named, and since $\vartheta$ ranges over an uncountable parameter space, that naming can be chosen to suit a problem. We show what this buys: in any base whose partial products absorb every denominator, a non-zero real number is rational precisely when it has a second name, and Cantor's 1869 irrationality criterion for Cantor series follows from the representation theory rather than from number theory.
Comments124 pages, 4 figures
