To illustrate how collections of events can fail to be collectives: births inthe Czech Republic are recorded at town halls of the local municipalitiesin large attractive cloth-bound books. Imagine the ministry in charge wantsto check the ratio of boys to girls over the last century, but does not want topay the cost of having someone tabulate 100 years‘ worth of paper records. - P8
More prosaically, seasonal events will not yield a collective. For example,
the frequency of flowering on a given day in Northern Europe is much higher in spring than in winter. So, picking out days in the spring months gives a higher relative frequency of blooms while picking out days in thewinter gives a lower relative frequency. (We could randomize this informa-tion by removing data about months, although why we would want to is beyond me.) - P8
1.2.3.1 Wald on collectives
To make our notion of randomness mathe-matically tractable, we have to have a function that picks out infinitesubsequences of collectives. We will call this function, following von Mises,
a place selection function. As we have seen, not just any place selectionfunction will do. First, we need a function that decides for each member ofa collective whether it is to be a member of the subsequence independently of what the value of the member happens to be. (For example every fifth member of a collective, or every member that follows four occurrences of some attribute like ‘lands on red‘.) - P9
1.2.3.2 Church‘s solution
In a footnote near the end of Church‘s 1940paper ‘The concept of a random sequence‘ he pointed out two problems with defining a theory of randomness in a given formal system L; first,
the definition of randomness will be relative to that language, and this,
to Church, seemed arbitrary; secondly, the notion of defining a gambling system relative to a logic leads to well-known problems:
It [Wald‘s interpretation of gambling systems] is unavoidably relative to the choice of the particular system L and thus has an element of arbitrariness which is artificial.
If used within the system L, it requires the presence in L of the semantical relationof denotation (known to be problematical on account of the Richard paradox). If itis used outside of L, it becomes necessary to say more exactly what is meant by
‘definable in L, and the questions of consistency and completeness of L are likely tobe raised in a peculiarly uncomfortable way.
(Church 1940: 135) - P10
We can think of recipes, especially those from Cook‘s Illustrated, as algorithms. Similarly,
a gambling system can be thought of as a recipe for success in the casino. - P10
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