Notes

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Several weak forms of \(AC\) are known to be equivalent to \(AC\) in \(ZF\)...

Content:

Several weak forms of AC are known to be equivalent to \(AC\) in \(ZF\). These include:

  1. \(MC\): The Axiom of Multiple Choice, Form 67
  2. \(LW\): Every linearly ordered set can be well ordered, Form 90;
  3. \(PW\): The power set of a well ordered set can be well ordered, Form 91;
  4. \(VB\): Every vector space has a basis, Form 66;
  5. \(A\): The Antichain Principle, Form 89; and
  6. \(C(LO,\infty)\): Every linearly ordered family of non-empty sets has a choice function, Form 202.

Consequently everything that implies one of these six forms is equivalent to \(AC\) in \(ZF\). That is, Form 95, Form 109, Form 112, Form 114, Form 133, Form 147, Form 149, Form 164, Form 218, Form 264, Form 292, Form 333, Form 334, and Form 335\((n)\) for \(n \ge 2\), also imply \(AC\) in \(ZF\). In fact in \(ZF^0\) the following two chains of implications are provable:

  • \(AC \to VB \to MC \to A \to LW \to PW\), and
  • \(AC \to C(LO,\infty)\to LW\).
With regard to the reverse implications, \(VB \to AC\), \(MC \to VB\) and \(LW \to C(LO,\infty)\) are open questions. The other reverse implications are known to be unprovable in \(ZF^0\). (See Jech [1973b] page 134, Felgner/Jech [1973], Blass [1984a], H. Rubin [1960], H. Rubin/J. Rubin [1985] Part I 5, Brunner [1983d], Brunner [1984b], Brunner [1985a], Howard/Rubin [1977], Truss [1978], Keremedis [1996a], Keremedis [1996b], and Note 18.) It follows from the two chains of implications that each of the twenty forms mentioned above implies \(PW\), Form 91, in \(ZF^0\).

In Pincus [1972a], page 740, it is shown that if \(\neg \Phi\) is boundable (see Pincus [1972a], p. 722 for the definition of boundable) and \(\Phi \rightarrow AC\) is a theorem of \(ZF\), then \(\Phi\rightarrow\) Form 91 \((PW)\) is a theorem of \(ZF^0\).

We also note that \(PW\) (Form 91) is equivalent to the axiom of choice for pure sets ([91 B]). To see this note first that [91 B] implies that the power set of an ordinal can be well ordered. Since every well ordered set is equivalent to an ordinal we obtain [91 B] \(\Rightarrow\) Form 91. For the other implication we use the result from H. Rubin/J. Rubin [1985], theorem 5.7, that Form 91 implies \(AC\) in \(ZF\).

Howard-Rubin number: 75

Type: Equivalencies

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