Description: 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.
Content:
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.
Let \(X = \{ y_i : i\in K\}\) be a family of finite sets. For each \(y_i\) let \(U_i\) be the real vector space \({\Bbb R}^{y_i}\) with pointwise addition and scalar multiplication. (If \(y_i = \{a_1,\ldots,a_n \}\) we could think of \(U_i\) as being the set of all formal sums \(k_1 a_1 + \cdots + k_n a_n\) where the \(k_i\)'s are real.) Let \(S_i\) be the subspace \(\{g\in U_i : g \mathrm{\; is \; constant}\}\). (Or in terms of formal sums, \(S_i\) is all formal sums \(k a_1 + \cdots + k a_n\).) Let \(V_i\) be the quotient space \(V_i = U_i/S_i\). That is, \(V_i\) consists of all equivalence classes \([g]\) of elements of \(U_i\) under the relation \(g\sim f \Leftrightarrow g-f\in S_i\). By form [1 DG] there is a family \(\{ F_i : i\in K\}\) such that for each \(i\in K\), \(F_i\) is an independent subset of \(V_i\). Since \(U_i\) is finite dimensional \(F_i\) must be finite. Say \(F_i = \{b_1,\ldots,b_r\}\) then since \(F_i\) is independent, the element \(w_i = b_1 + \cdots + b_r\) is not zero. The vector \(w_i = [g]\) for some \(g\in U_i\). Assume \(f\in [g]\). Then if \(a,a'\in y_i\) and \(g(a)\le g(a')\) it follows from the fact that \(f - g\) is constant that \(f(a) \le f(a')\). Therefore the set \(K_i = \{ a\in y_i : g(a)\) is minimum among the numbers \(g(a')\) for \(a'\in y_i \}\) is independent of the choice of \(g\in w_i\). It is also true that \(K_i \ne y_i\). This follows from the fact that \(w_i \ne 0\) which implies that \(g\) is not constant. The family \(\{K_i : i\in K \}\) is therefore a Kinna-Wagner function for \(X\).
Howard-Rubin number: 127
Type: proof of equivalencies
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