Axiom Of Choice

We have the following indirect implication of form equivalence classes:

258 $\Rightarrow$ 106 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
258 $\Rightarrow$ 255	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
255 $\Rightarrow$ 260	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
260 $\Rightarrow$ 40	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
40 $\Rightarrow$ 43	Consistency results for $ZF$, Jensen, R.B. 1967, Notices Amer. Math. Soc. On cardinals and their successors, Jech, T. 1966a, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
43 $\Rightarrow$ 106	Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc. On the role of the Baire category theorem and dependent choice in the foundations of logic, Goldblatt, R. 1985, J. Symbolic Logic

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

Howard-Rubin Number	Statement
258:	$Z(D,L)$: Every directed relation $(X,R)$ in which linearly ordered subsets have upper bounds, has a maximal element.
255:	$Z(D,R)$: Every directed relation $(P,R)$ in which every ramified subset $A$ has an upper bound, has a maximal element.
260:	$Z(TR\&C,P)$: If $(X,R)$ is a transitive and connected relation in which every partially ordered subset has an upper bound, then $(X,R)$ has a maximal element.
40:	$C(WO,\infty)$: Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.
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.
106:	Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire.

Comment:

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Implication	Reference
258 \(\Rightarrow\) 255	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
255 \(\Rightarrow\) 260	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
260 \(\Rightarrow\) 40	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
40 \(\Rightarrow\) 43	Consistency results for $ZF$, Jensen, R.B. 1967, Notices Amer. Math. Soc. On cardinals and their successors, Jech, T. 1966a, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
43 \(\Rightarrow\) 106	Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc. On the role of the Baire category theorem and dependent choice in the foundations of logic, Goldblatt, R. 1985, J. Symbolic Logic

Howard-Rubin Number	Statement
258:	\(Z(D,L)\): Every directed relation \((X,R)\) in which linearly ordered subsets have upper bounds, has a maximal element.
255:	\(Z(D,R)\): Every directed relation \((P,R)\) in which every ramified subset \(A\) has an upper bound, has a maximal element.
260:	\(Z(TR\&C,P)\): If \((X,R)\) is a transitive and connected relation in which every partially ordered subset has an upper bound, then \((X,R)\) has a maximal element.
40:	\(C(WO,\infty)\): Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.
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.
106:	Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire.