Axiom Of Choice

We have the following indirect implication of form equivalence classes:

359 $\Rightarrow$ 78 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 $\Rightarrow$ 78	The Axiom of Choice, Jech, [1973b] The Axiom of Choice, Jech, [1973b]

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.
78:	Urysohn's Lemma: If $A$ and $B$ are disjoint closed sets in a normal space $S$, then there is a continuous $f:S\rightarrow [0,1]$ which is 1 everywhere in $A$ and 0 everywhere in $B$. Urysohn [1925], pp 290-292.

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 \(\Rightarrow\) 78	The Axiom of Choice, Jech, [1973b] The Axiom of Choice, Jech, [1973b]

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.
78:	Urysohn's Lemma: If \(A\) and \(B\) are disjoint closed sets in a normal space \(S\), then there is a continuous \(f:S\rightarrow [0,1]\) which is 1 everywhere in \(A\) and 0 everywhere in \(B\). Urysohn [1925], pp 290-292.