Implications

We have the following indirect implication of form equivalence classes:

43 \(\Rightarrow\) 42
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
43 \(\Rightarrow\) 42 clear

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

Howard-Rubin Number Statement
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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