Axiom Of Choice

We have the following indirect implication of form equivalence classes:

20 $\Rightarrow$ 412 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
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 $\Rightarrow$ 411	Dependent Choice and Weak Compactness, Delhomme-Morillon-2000[2000], Notre Dame J. Formal Logic
411 $\Rightarrow$ 412	clear

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

Howard-Rubin Number	Statement
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.
411:	RCuc (Reflexive Compactness for uniformly convex Banach spaces): The closed unit ball of a uniformly convex Banach space is compact for the weak topology.
412:	RCh (Reflexive Compactness for Hilbert spaces): The closed unit ball of a Hilbert space is compact for the weak topology.

Comment:

Back

Implication	Reference
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 \(\Rightarrow\) 411	Dependent Choice and Weak Compactness, Delhomme-Morillon-2000[2000], Notre Dame J. Formal Logic
411 \(\Rightarrow\) 412	clear

Howard-Rubin Number	Statement
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.
411:	RCuc (Reflexive Compactness for uniformly convex Banach spaces): The closed unit ball of a uniformly convex Banach space is compact for the weak topology.
412:	RCh (Reflexive Compactness for Hilbert spaces): The closed unit ball of a Hilbert space is compact for the weak topology.