Axiom Of Choice

We have the following indirect implication of form equivalence classes:

101 $\Rightarrow$ 211 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
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 $\Rightarrow$ 211	clear

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

Howard-Rubin Number	Statement
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.
211:	$DCR$: Dependent choice for relations on $\Bbb R$: If $R\subseteq\Bbb R\times\Bbb R$ satisfies $(\forall x\in \Bbb R)(\exists y\in\Bbb R)(x\mathrel R y)$ then there is a sequence $\langle x(n): n\in\omega\rangle$ of real numbers such that $(\forall n\in\omega)(x(n)\mathrel R x(n+1))$.

Comment:

Back

Implication	Reference
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 \(\Rightarrow\) 211	clear

Howard-Rubin Number	Statement
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.
211:	\(DCR\): Dependent choice for relations on \(\Bbb R\): If \(R\subseteq\Bbb R\times\Bbb R\) satisfies \((\forall x\in \Bbb R)(\exists y\in\Bbb R)(x\mathrel R y)\) then there is a sequence \(\langle x(n): n\in\omega\rangle\) of real numbers such that \((\forall n\in\omega)(x(n)\mathrel R x(n+1))\).