Axiom Of Choice

We have the following indirect implication of form equivalence classes:

347 $\Rightarrow$ 388 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
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 $\Rightarrow$ 388

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

Howard-Rubin Number	Statement
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.
388:	Every infinite branching poset (a partially ordered set in which each element has at least two lower bounds) has either an infinite chain or an infinite antichain.

Comment:

Back

Implication	Reference
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 \(\Rightarrow\) 388

Howard-Rubin Number	Statement
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.
388:	Every infinite branching poset (a partially ordered set in which each element has at least two lower bounds) has either an infinite chain or an infinite antichain.