Implications

We have the following indirect implication of form equivalence classes:

2 \(\Rightarrow\) 186
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
2 \(\Rightarrow\) 3 On successors in cardinal arithmetic, Truss, J. K. 1973c, Fund. Math.
3 \(\Rightarrow\) 186 Set theory for the mathematician, Rubin, [1967]

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

Howard-Rubin Number Statement
2:

Existence of successor cardinals: For every cardinal \(m\) there is a cardinal \(n\) such that \(m < n\) and \((\forall p < n)(p \le m)\).

3:  \(2m = m\): For all infinite cardinals \(m\), \(2m = m\).
186:

Every pair of cardinal numbers has a least upper bound (in the usual cardinal ordering.)

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