Since the Intuitionistic school of mathematics, as formulated by L. E. J. Brouwer, rejects the idea of a completed infinity, in order to use a sequence (which is, in classical mathematics, an infinite object), we must have a formulation of a finite, constructible object that can serve the same purpose as a sequence.
Thus, Brouwer formulated the choice sequence, which is given as a construction, rather than an abstract, infinite object.
[1] A distinction is made between lawless and lawlike sequences.
can be thought of as a lawlike sequence: the sequence can be fully constructively described by the unique element 0 and a successor 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
) such that: 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
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.
There are two axioms in particular that we expect to hold of choice sequences as described above.
will probably be an integer encoding a finite initial sequence).
We expect the following, called the axiom of open data, to hold of all lawless sequences:
The intuitive justification for this axiom is as follows: in intuitionistic mathematics, verification that
is given as a procedure; at any point of execution of this procedure, we will have examined only a finite initial segment of the sequence.
Intuitively, then, this axiom states that since, at any point of verifying that
holds for a finite initial sequence of
that we have already examined, if we run the identical procedure on
The axiom can be generalized for any predicate taking an arbitrary number of arguments.
Another axiom is required for lawless sequences.
states that, for any finite prefix (encoded by)
We require this axiom so as not to have any "holes" in the set of choice sequences.
This axiom is the reason we require that arbitrarily long finite initial sequences of lawless choice sequences can be specified in advance; without this requirement, the axiom of density is not necessarily guaranteed.