Axiom Of Choice

We have the following indirect implication of form equivalence classes:

101 $\Rightarrow$ 78 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 $\Rightarrow$ 78	The Axiom of Choice, Jech, [1973b] The Axiom of Choice, Jech, [1973b]

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.
78:	Urysohn's Lemma: If $A$ and $B$ are disjoint closed sets in a normal space $S$, then there is a continuous $f:S\rightarrow [0,1]$ which is 1 everywhere in $A$ and 0 everywhere in $B$. Urysohn [1925], pp 290-292.

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 \(\Rightarrow\) 78	The Axiom of Choice, Jech, [1973b] The Axiom of Choice, Jech, [1973b]

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.
78:	Urysohn's Lemma: If \(A\) and \(B\) are disjoint closed sets in a normal space \(S\), then there is a continuous \(f:S\rightarrow [0,1]\) which is 1 everywhere in \(A\) and 0 everywhere in \(B\). Urysohn [1925], pp 290-292.