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

TL;DR

This paper constructs a countable, dense, Dedekind-complete subset of the real numbers by extending rationals with irrational markings, challenging traditional views on uncountability and the continuum hypothesis.

Contribution

It presents explicit constructions of countable sets with properties usually attributed only to the continuum, highlighting model-dependent aspects of set cardinality.

Findings

Constructed a countable dense Dedekind-complete set within ZFC.

Built a similar set within Wang's $\\Sigma$-model using diagonalization.

Revealed foundational tensions between classical non-denumerability proofs and the Nested Interval Property.

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.