Choice Sequence - Lawlike and Lawless Sequences

Lawlike and Lawless Sequences

A distinction is made between lawless and lawlike sequences. A lawlike sequence is one that can be described completely — it is a completed construction, that can be fully described. For example, the natural numbers can be thought of as a lawlike sequence: the sequence can be fully constructively described by the unique element 0 and a successor function. Given this formulation, we know that the th element in the sequence of natural numbers will be the number . Similarly, a function mapping from the natural numbers into the natural numbers effectively determines the value for any argument it takes, and thus describes a lawlike sequence.

A lawless (also, free) sequence, on the other hand, is one that is not predetermined. It is to be thought of as a procedure for generating values for the arguments 0, 1, 2, .... That is, a lawless sequence is a procedure for generating, ... (the elements of the sequence ) such that:

• At any given moment of construction of the sequence, only an initial segment of the sequence is known, and no restrictions are placed on the future values of ; and
• One may specify, in advance, an initial segment of .

Note that the first point above is slightly misleading, as we may specify, for example, that the values in a sequence be drawn exclusively from the set of natural numbers—we can specify, a priori, the range of the sequence.

The canonical example of a lawless sequence is the series of rolls of a die. We specify which die to use and, optionally, specify in advance the values of the first rolls (for ). Further, we restrict the values of the sequence to be in the set . This specification comprises the procedure for generating the lawless sequence in question. At no point, then, is any particular future value of the sequence known.