 The failure of the other direction, the implication from right to left, is due to the possibility that \(\alpha\) and \(\beta\) are of different types, so that any similarity relation between them will have its domain and range in different types. Suppose there are \(\aleph_0\) individuals, and consider two higher types with the cardinals in them of even larger, but distinct, cardinalities, say \(\alpha\) in some high type has cardinality \(\aleph_1\) and that \(\beta\) in an even higher type and has cardinality \(\aleph_2\). There are no sets of individuals similar to \(\alpha\) or \(\beta\), so no similarity relation with a domain in \(\alpha\) or \(\beta\) will hold with respect to any set in the type of individuals. Suppose that \(\#\) is defined in terms of such descending relation. Therefore \(\# \alpha = \{ \Lambda \} = 0\) and \(\# \beta = \{ \Lambda \} = 0\) so \(\# \alpha = \# \beta\), yet \(\alpha \not\approx \beta\) on any similarity relation of whatever domain and range, because their cardinalities differ. Whitehead and Russell assert that the case of \(\alpha\) and \(\beta\) being in different types is the only way to construct an exception to this direction of Hume’s principle, and offer as a restricted version: