Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
192 \(\Rightarrow\) 43	Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.
43 \(\Rightarrow\) 42	clear

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

Howard-Rubin Number	Statement
192:	\(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\).
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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