 Premise 1 is beyond reproach. However, premises 2, 3, and 4 require us to be careful about how lengths add. Recall that in section  3.2 we discussed how to add a countable sequence of numbers – but the method described there depends on the order, and requires a countable, well-ordered sequence. Although there are techniques for summing uncountable sets of non-negative numbers, most mathematicians deny that lengths or other measures can be added in these ways. This is parallel to what Kolmogorov said about probability (see section  6.3).  Probability and length are two paradigms of the more general mathematical field of “measure theory”, which includes all such countably additive real-valued functions. For a more detailed discussion of this problem, including approaches involving infinitesimal length, see Skyrms (1983).