Implications

We have the following indirect implication of form equivalence classes:

91 \(\Rightarrow\) 35
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
91 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 94 clear
94 \(\Rightarrow\) 35 Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.

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

Howard-Rubin Number Statement
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.

94:

\(C(\aleph_{0},\infty,{\Bbb R})\): Every denumerable family of non-empty sets of reals  has a choice function. Jech [1973b], p 148 prob 10.1.

35:

The union of countably many meager subsets of \({\Bbb R}\) is meager. (Meager sets are the same as sets of the first category.) Jech [1973b] p 7 prob 1.7.

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