Axiom Of Choice

We have the following indirect implication of form equivalence classes:

168 $\Rightarrow$ 411 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
168 $\Rightarrow$ 100	clear
100 $\Rightarrow$ 347	Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic
347 $\Rightarrow$ 40	Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, 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 $\Rightarrow$ 411	Dependent Choice and Weak Compactness, Delhomme-Morillon-2000[2000], Notre Dame J. Formal Logic

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

Howard-Rubin Number	Statement
168:	Dual Cantor-Bernstein Theorem:$(\forall x) (\forall y)(\|x\| \le^\|y\|$ and $\|y\|\le^ \|x\|$ implies $\|x\| = \|y\|)$ .
100:	Weak Partition Principle: For all sets $x$ and $y$, if $x\precsim^* y$, then it is not the case that $y\prec x$.
347:	Idemmultiple Partition Principle: If $y$ is idemmultiple ($2\times y\approx y$) and $x\precsim ^* y$, then $x\precsim y$.
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.

Comment:

Back

Implication	Reference
168 \(\Rightarrow\) 100	clear
100 \(\Rightarrow\) 347	Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic
347 \(\Rightarrow\) 40	Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, 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 \(\Rightarrow\) 411	Dependent Choice and Weak Compactness, Delhomme-Morillon-2000[2000], Notre Dame J. Formal Logic

Howard-Rubin Number	Statement
168:	Dual Cantor-Bernstein Theorem:\((\forall x) (\forall y)(\|x\| \le^\|y\|\) and \(\|y\|\le^ \|x\|\) implies \(\|x\| = \|y\|)\) .
100:	Weak Partition Principle: For all sets \(x\) and \(y\), if \(x\precsim^* y\), then it is not the case that \(y\prec x\).
347:	Idemmultiple Partition Principle: If \(y\) is idemmultiple (\(2\times y\approx y\)) and \(x\precsim ^* y\), then \(x\precsim y\).
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.