Axiom Of Choice

We have the following indirect implication of form equivalence classes:

168 $\Rightarrow$ 42 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 $\Rightarrow$ 42	clear

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.
42:	Löwenheim-Skolem Theorem: If a countable family of first order sentences is satisfiable in a set $M$ then it is satisfiable in a countable subset of $M$. (See Moore, G. [1982], p. 251 for references.

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 \(\Rightarrow\) 42	clear

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.
42:	Löwenheim-Skolem Theorem: If a countable family of first order sentences is satisfiable in a set \(M\) then it is satisfiable in a countable subset of \(M\). (See Moore, G. [1982], p. 251 for references.