A Countable, Dense, Dedekind-Complete Subset of $\mathbb{R}$ Constructed by Extending $\mathbb{Q}$ via Simultaneous Marking of Closed Intervals with Rational Endpoints

Abstract

This article explores the model-dependent nature of set cardinality, emphasizing that cardinality is not absolute but varies across different axiomatic frameworks. Although Cantor's diagonal argument shows the real numbers are non-denumerable within ZF (Zermelo-Fraenkel set theory), the precise cardinality of the continuum remains unsettled and depends critically on model assumptions. For instance, under G\"odel's inner-model axiom V=Ultimate L, the Continuum Hypothesis (CH) holds, whereas Martin's Axiom implies its negation. The L\"owenheim-Skolem theorem further illustrates this relativity by demonstrating that any first-order theory admitting a non-denumerable model must also admit denumerable models, highlighting that even the notions of "denumerable" and "non-denumerable" are inherently model-relative. To examine these issues concretely, we construct two countable sets with properties typically attributed only to the continuum. First, within ZFC (ZF plus Axiom of Choice), we build a countable set $S_m$ from all closed intervals with rational endpoints. By assigning irrational marks simultaneously to each interval, respecting the nested interval structure, we obtain a set that is everywhere dense and Dedekind complete, yet countable. Next, we explicitly construct a similar set within Wang's $\Sigma$-model by systematically inserting irrational numbers between rational numbers via infinite diagonalization, resulting in a constructive enumeration of reals. These findings identify foundational tensions between classical proofs of non-denumerability and the Nested Interval Property, prompting a reevaluation of cardinality and CH within formal set theory.