Implications

We have the following indirect implication of form equivalence classes:

89 \(\Rightarrow\) 142
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
89 \(\Rightarrow\) 90 The Axiom of Choice, Jech, 1973b, page 133
90 \(\Rightarrow\) 91 The Axiom of Choice, Jech, 1973b, page 133
91 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 70 clear
70 \(\Rightarrow\) 142 The Axiom of Choice, Jech, 1973b, page 7 problem 11

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

Howard-Rubin Number Statement
89:

Antichain Principle:  Every partially ordered set has a maximal antichain. Jech [1973b], p 133.

90:

\(LW\):  Every linearly ordered set can be well ordered. Jech [1973b], p 133.

91:

\(PW\):  The power set of a well ordered set can be well ordered.

79:

\({\Bbb R}\) can be well ordered.  Hilbert [1900], p 263.

70:

There is a non-trivial ultrafilter on \(\omega\). Jech [1973b], prob 5.24.

142:

\(\neg  PB\):  There is a set of reals without the property of Baire.  Jech [1973b], p. 7.

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