False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (openai/gpt-5.4)audited 2026-07-24

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

FALSE: some rational number squares to 2

Statement

False claim: there exists $x \in \mathbb{Q}$ with $x^2 = 2$ .

Facts & Assumptions

Given: The rationals as quotients of integer pairs (The rationals as equivalence classes of pairs of integers).

[A1]

Every rational can be written $a/b$ in lowest terms, with $a, b$ not both even (cancel common factors of $2$ ; the process terminates by well-ordering of $\mathbb{N}$ ).

[A2]

Parity in $\mathbb{Z}$ : if $k^2$ is even then $k$ is even (an odd $k = 2j+1$ has odd square $4j^2 + 4j + 1$ ).

Refutation

technique · contradiction

1.1

Assume, for contradiction, that some $x \in \mathbb{Q}$ has $x^2 = 2$ ; write $x = a/b$ in lowest terms, $a, b$ not both even.

assume-contraA1

2.1

Clearing denominators: $a^2 = 2b^2$ .

step 1.1algebra

3.1

$a^2$ is even, hence $a$ is even: $a = 2c$ .

step 2.1A2

4.1

Substituting: $4c^2 = 2b^2$ , so $b^2 = 2c^2$ is even, hence $b$ is even.

step 3.1A2algebra

5.1

Both $a$ and $b$ are even, contradicting the lowest-terms choice of step 1.1; the assumption fails: no rational squares to $2$ .

step 4.1step 1.1discharge-contradiction∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 20 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

T. Tao, Analysis I, 3rd ed., §4.4 (standard reference, not scraped)
Square root of 2 — proofs of irrationality (Wikipedia) (standard reference, not scraped)