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

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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FALSE: there exist two non-isomorphic complete ordered fields

Statement

False statement. There exist two complete ordered fields (Complete ordered field (least-upper-bound property)) that are not isomorphic; that is, completeness together with the ordered-field axioms fails to determine the real numbers up to isomorphism.

Facts & Assumptions

Given: The claim that there are complete ordered fields $F$ and $G$ admitting no ordered-field isomorphism between them.

[L1]

Any two complete ordered fields are isomorphic via a unique ordered-field isomorphism (Uniqueness of the complete ordered field: $R$ up to a unique isomorphism).

[L2]

A complete ordered field is one with the least-upper-bound property (Complete ordered field (least-upper-bound property)).

[L3]

An ordered-field isomorphism is a bijective, order-preserving field homomorphism (Ordered-field isomorphism).

Refutation

technique · direct

1.1

Let $F$ and $G$ be any complete ordered fields, as the claim posits, each with the least-upper-bound property.

givenL2

1.2

By [L1] there exists an ordered-field isomorphism $φ : F \to G$ (indeed a unique one).

2.1

Hence $F$ and $G$ are isomorphic, contradicting the asserted non-isomorphism; since $F, G$ were arbitrary complete ordered fields, no non-isomorphic pair can exist and the statement is false.

step 1.1step 1.2L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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 (standard reference, not scraped)
M. Spivak, Calculus, 4th ed., Ch. 30 (standard reference, not scraped)