Implications

We have the following indirect implication of form equivalence classes:

363 \(\Rightarrow\) 170
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
363 \(\Rightarrow\) 368 clear
368 \(\Rightarrow\) 170 On well-ordered subsets of any set, Tarski, A. 1939, Fund. Math.

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

Howard-Rubin Number Statement
363:

There are exactly \(2^{\aleph_0}\) Borel sets in \(\Bbb R\). G. Moore [1982], p 325.

368:

The set of all denumerable subsets of \(\Bbb R\) has power \(2^{\aleph_0}\).

170:

\(\aleph_{1}\le 2^{\aleph_{0}}\).

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