 Critics fail to be convinced by these paradoxes of infinity. For example, Rundle (2004: 170) agrees with Craig that the concept of an actual infinite is paradoxical, but this, he argues, provides no grounds for thinking it is incoherent. The logical problems with the actual infinite are not problems of incoherence but arise from the features that are characteristic of infinite sets. When the intuitive notion of “smaller than” is replaced by a precise definition, finite sets and infinite sets just behave somewhat differently, that is all. Cantor and all subsequent set theorists define a set \(B\) to be smaller than set \(A\) (i.e., has fewer members) just in case \(B\) is the same size as a subset of \(A\), but \(A\) is not the same size as any subset of \(B\). The application of this definition to finite and infinite sets yields results that Craig finds counter-intuitive but which mathematicians see as our best understanding for comparing the size of sets. They see the fact that an infinite set can be put into one-to-one correspondence with one of its own proper subsets as one of the defining characteristics of an infinite set, not an absurdity. Say that set \(C\) is a proper subset of \(A\) just in case every element of \(C\) is an element of \(A\) while \(A\) has some element that is not an element of \(C\). In finite sets, but not necessarily in infinite sets, when set \(B\) is a proper subset of \(A\), \(B\) is smaller than \(A\). However, this does not necessarily hold for infinite sets—as above where \(B\) is the set of squares of natural numbers and \(A\) is the set of all natural numbers. The proper subset \(B\) might be “smaller” than \(A\). What is crucial is that \(B\) is not smaller in the sense of having a smaller number of members than \(A\) (i.e., a smaller cardinality).