Implications

We have the following indirect implication of form equivalence classes:

130 \(\Rightarrow\) 137-k
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
130 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 139
139 \(\Rightarrow\) 137-k Cancellation laws for surjective cardinals, Truss, J. K. 1984, Ann. Pure Appl. Logic

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

Howard-Rubin Number Statement
130:

\({\cal P}(\Bbb R)\) is well orderable.

79:

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

139:

Using the discrete topology on 2, \(2^{\cal P(\omega)}\) is compact.

137-k:

Suppose \(k\in\omega-\{0\}\). If \(f\) is a 1-1 map from \(k\times X\) into \(k\times Y\) then there are partitions \(X = \bigcup_{i \le k} X_{i} \) and \(Y = \bigcup_{i \le k} Y_{i} \) of \(X\) and \(Y\) such that \(f\) maps \(\bigcup_{i \le k} (\{i\} \times  X_{i})\) onto \(\bigcup_{i \le k} (\{i\} \times  Y_{i})\).

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