Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-01
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FALSE: every inverse limit of surjective finite-group systems has surjective coordinate projections in ZF

Statement

In ZF, every inverse system of finite groups with surjective transition maps has surjective coordinate projections from its inverse limit.

Facts & Assumptions

Given: The classical set-theoretic fact that this surjectivity principle is not provable in ZF for arbitrary infinite inverse systems of finite groups.

[L1]

The coordinate projections from an inverse limit are the canonical maps, and the universal property alone does not assert their surjectivity (The inverse limit has its canonical coordinate projection maps, The compatible-tuple construction satisfies the inverse-limit universal property in groups).

Refutation

technique · direct
1.1

The statement is a global set-theoretic assertion about all surjective inverse systems, not a theorem of ZF proved by [L1]. In models of ZF without sufficient choice, there are inverse systems of finite groups with surjective bonding maps for which a prescribed coordinate value has no compatible lift.

givenL1
2.1

In such a model, the corresponding coordinate projection from the inverse limit fails to be surjective. Therefore the universal claim in the Statement is false in ZF.

step 1.1
3.1

So surjectivity of all coordinate projections requires extra choice beyond bare ZF. This refutes the statement.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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