Implications

We have the following indirect implication of form equivalence classes:

44 \(\Rightarrow\) 412
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
44 \(\Rightarrow\) 43 The interdependence of certain consequences of the axiom of choice, Levy, A. 1964, Fund. Math.
The Axiom of Choice, Jech, 1973b, page 120 theorem 8.1
43 \(\Rightarrow\) 411 Dependent Choice and Weak Compactness, Delhomme-Morillon-2000[2000], Notre Dame J. Formal Logic
411 \(\Rightarrow\) 412 clear

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
44:

\(DC(\aleph _{1})\):  Given a relation \(R\) such that for every  subset \(Y\) of a set \(X\) with \(|Y| < \aleph_{1}\) there is an \(x \in  X\)  with \(Y \mathrel R x\), then there is a function \(f: \aleph_{1} \rightarrow  X\) such that \((\forall\beta < \aleph_{1}) (\{f(\gamma ): \gamma < b \} \mathrel R f(\beta))\).

43:

\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\)  is  a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\)  then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\).  See Tarski [1948], p 96, Levy [1964], p. 136.

411:

RCuc (Reflexive Compactness for uniformly convex Banach spaces): The closed unit ball of a uniformly convex Banach space is compact for the weak topology.

412:

RCh (Reflexive Compactness for Hilbert spaces): The closed unit ball of a Hilbert space is compact for the weak topology.

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