 Suppose again that \(A\) is the proposition Heads on toss 1 and that the propositions in \(\mathcal{F}\) that are relevantly similar to \(A\) according to \(\sim\) are Heads on toss 1, …, Heads on toss 1000. Now suppose that \(\mathcal{F}_1\) extends \(\mathcal{F}\) by introducing a new proposition about a further hypothetical toss of the coin (as well as perhaps other propositions). That is, it introduces Heads on toss 1001 (and closes out under negation, disjunction, and conjunction). And suppose that \(\sim_1\) extends \(\sim\), so that the new proposition Heads on toss 1001 is considered relevantly similar to each Heads on toss 1, …, Heads on toss 1000. Then those who appeal to hypothetical limiting frequencies must claim that there is a unique number that gives what the frequency of heads would be, were the coin tossed 1001 times. They denote this number \(\mathrm{Freq}(\mathcal{F}_1, A, \sim_1, w)\). Now suppose that \(\mathcal{F}_2\) extends \(\mathcal{F}_1\) by adding the new proposition Heads on toss 1002 and \(\sim_2\) extends \(\sim_1\), so that the new proposition Heads on toss 1002 is considered relevantly similar to each Heads on toss 1, …, Heads on toss 1001. And so on. Then the limiting relative frequency of \(A\) at \(w\) (written \(\mathrm{LimFreq}(\mathcal{F}, A, \sim, w)\)) is the number towards which the following sequence tends:   \[\mathrm{Freq}(\mathcal{F}, A, \sim, w), \mathrm{Freq}(\mathcal{F}_1, A, \sim_1, w), \mathrm{Freq}(\mathcal{F}_2, A, \sim_2, w), \ldots\]