Implications

We have the following indirect implication of form equivalence classes:

192 \(\Rightarrow\) 16
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 \(\Rightarrow\) 8 clear
8 \(\Rightarrow\) 16 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.

8:

\(C(\aleph_{0},\infty)\):

16:

\(C(\aleph_{0},\le 2^{\aleph_{0}})\):  Every denumerable collection of non-empty sets  each with power \(\le  2^{\aleph_{0}}\) has a choice function.

Comment:

Back