Axiom Of Choice

We have the following indirect implication of form equivalence classes:

44 \(\Rightarrow\) 42 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 \(\Rightarrow\) 42	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.
42:	Löwenheim-Skolem Theorem: If a countable family of first order sentences is satisfiable in a set \(M\) then it is satisfiable in a countable subset of \(M\). (See Moore, G. [1982], p. 251 for references.

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