Form equivalence class Howard-Rubin Number: 1
Statement:
Vector Space Multiple Choice: For every family \(V = \{\,V_i : i \in K\,\}\) of non-trivial vector spaces there is a family \(F = \{\, F_i : i\in K\,\}\) such that for each \(i\in K\),\(F_i\) is a non-empty, finite, independent subset of \(V_i\).
Howard-Rubin number: 1 BZ
Citations (articles):
Connections (notes):
Note [127]
Forms [1 BZ] (Vector space multiple
choice) and [1 DG] (Vector space Kinna-Wagner principle) were
suggested by K. Keremedis. It is clear that [1 BZ] implies Form 346. In this note we prove that [1 DG] implies the
Kinna-Wagner principle \(KW(\infty,< \aleph_0)\) (form [62 E]).
Since the axiom of choice is implied by the conjunction of forms
Form 62 and Form 67, we obtain a proof that [1 DG]
+ Form 67 implies the axiom of
choice. (Form Form 62 is \(C(\infty,<\aleph_0)\) and Form 67
is the axiom o7 multiple choice.) Keremedis [1999d] proves
that [1 DG]
implies Form 67 to complete the proof that [1 DG] implies the axiom
of choice. Similarly, since [1 BZ] implies Form 67, we obtain the
result: [1 BZ] implies the axiom of choice.
References (books):
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