False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (openai/gpt-5.4)audited 2026-07-25

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FALSE: every Dedekind cut has a greatest element

Statement

False claim: every Dedekind cut (Dedekind cut) has a greatest element: some $m \in A$ with $q \leq m$ for all $q \in A$ .

Facts & Assumptions

Given: The cut axioms (C1)–(C3) (Dedekind cut), $R$ as the set of all cuts (The real numbers $R$ as Dedekind cuts), and the rational cut $0^{*} : = {q \in Q : q < 0}$ .

[L1]

$Q$ is a totally ordered field; in particular for $q < 0$ the rational $q / 2$ satisfies $q < q / 2 < 0$ (The rationals form a totally ordered field).

Refutation

technique · direct

1.1givenL1

(C1, C2) $0^{*} = {q \in Q : q < 0}$ is nonempty ( $- 1 \in 0^{*}$ ) and proper ( $0 \notin 0^{*}$ ), and it is downward closed: $q < p < 0 \Rightarrow q < 0$ , so $q \in 0^{*}$ .

1.2givenL1

(C3) For any $q \in 0^{*}$ , i.e. $q < 0$ , the rational $q / 2$ satisfies $q < q / 2 < 0$ , so $q / 2 \in 0^{*}$ and $q / 2 > q$ ; hence no element of $0^{*}$ is greatest.

2.1step 1.1step 1.2

By steps 1.1 and 1.2, $0^{*}$ satisfies (C1)–(C3): it is a Dedekind cut (Dedekind cut), i.e. a real number (The real numbers $R$ as Dedekind cuts), with no greatest element.

3.1step 2.1given∎

Thus $0^{*}$ is a Dedekind cut having no greatest element, directly contradicting the claim; more strongly, "no greatest element" is precisely axiom (C3), which every cut must satisfy, so the claim fails for all cuts and is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1 (Appendix: construction of ℝ) (standard reference, not scraped)
Math 331 course handout: Dedekind Cuts and Real Numbers (Hobart and William Smith Colleges) (standard reference, not scraped)
Dedekind cut (Wikipedia) (standard reference, not scraped)