Axiom Of Choice

We have the following indirect implication of form equivalence classes:

359 $\Rightarrow$ 106 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
359 $\Rightarrow$ 20	clear
20 $\Rightarrow$ 101	Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic
101 $\Rightarrow$ 40	On some weak forms of the axiom of choice in set theory, Pelc, A. 1978, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
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
359:	If $\{A_{x}: x\in S\}$ and $\{B_{x}: x\in S\}$ are families of pairwise disjoint sets and $ \|A_{x}\| \le \|B_{x}\|$ for all $x\in S$, then $\|\bigcup_{x\in S}A_{x}\| \le \|\bigcup_{x\in S} B_{x}\|$.
20:	If $\{A_{x}: x \in S \}$ and $\{B_{x}: x \in S\}$ are families of pairwise disjoint sets and $ \|A_{x}\| = \|B_{x}\|$ for all $x\in S$, then $\|\bigcup_{x\in S}A_{x}\| = \|\bigcup_{x\in S} B_{x}\|$. Moore [1982] (1.4.12 and 1.7.8).
101:	Partition Principle: If $S$ is a partition of $M$, then $S \precsim M$.
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
359 \(\Rightarrow\) 20	clear
20 \(\Rightarrow\) 101	Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic
101 \(\Rightarrow\) 40	On some weak forms of the axiom of choice in set theory, Pelc, A. 1978, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
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
359:	If \(\{A_{x}: x\in S\}\) and \(\{B_{x}: x\in S\}\) are families of pairwise disjoint sets and \( \|A_{x}\| \le \|B_{x}\|\) for all \(x\in S\), then \(\|\bigcup_{x\in S}A_{x}\| \le \|\bigcup_{x\in S} B_{x}\|\).
20:	If \(\{A_{x}: x \in S \}\) and \(\{B_{x}: x \in S\}\) are families of pairwise disjoint sets and \( \|A_{x}\| = \|B_{x}\|\) for all \(x\in S\), then \(\|\bigcup_{x\in S}A_{x}\| = \|\bigcup_{x\in S} B_{x}\|\). Moore [1982] (1.4.12 and 1.7.8).
101:	Partition Principle: If \(S\) is a partition of \(M\), then \(S \precsim M\).
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.