In
intuitionistic mathematics, a choice sequence is a
constructive
Although the general English usage of the adjective constructive is "helping to develop or improve something; helpful to someone, instead of upsetting and negative," as in the phrase "constructive criticism," in legal writing ''constructive'' has ...
formulation of a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
. Since the Intuitionistic school of mathematics, as formulated by
L. E. J. Brouwer
Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and compl ...
, 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.
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
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
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
("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ...
, 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.
Axiomatization
There are two
axioms in particular that we expect to hold of choice sequences as described above. Let
denote the relation "the sequence
begins with the initial sequence
" for choice sequence
and finite segment
(more specifically,
will probably be an integer
encoding
In communications and information processing, code is a system of rules to convert information—such as a letter, word, sound, image, or gesture—into another form, sometimes shortened or secret, for communication through a communication ...
a finite initial sequence).
We expect the following, called the ''axiom of open data'', to hold of all lawless sequences:
where_
_is_a_
alpha\in_n_\,\land\,_\forall\beta\in_n[A(\beta)
where_
_is_a_one-place_predicate">(\beta).html"_;"title="alpha\in_n_\,\land\,_\forall\beta\in_n[A(\beta)">alpha\in_n_\,\land\,_\forall\beta\in_n[A(\beta)
where_
_is_a_one-place_predicate.__The_intuitive_justification_for_this_axiom_is_as_follows:_in_intuitionistic_mathematics,_verification_that_
_holds_of_the_sequence_
_is_given_as_a_algorithm.html" ;"title="one-place_predicate.html" ;"title="(\beta).html" ;"title="alpha\in n \,\land\, \forall\beta\in n[A(\beta)">alpha\in n \,\land\, \forall\beta\in n[A(\beta)
where
is a one-place predicate">(\beta).html" ;"title="alpha\in n \,\land\, \forall\beta\in n[A(\beta)">alpha\in n \,\land\, \forall\beta\in n[A(\beta)
where
is a one-place predicate. The intuitive justification for this axiom is as follows: in intuitionistic mathematics, verification that
holds of the sequence
is given as a algorithm">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 of
, we will only have verified that
holds for a finite initial sequence of
; thus, it must be the case that
also holds for any lawless sequence
sharing this initial sequence. This is so because, at any point in the procedure of verifying
, for any such
sharing the initial prefix of
encoded by
that we have already examined, if we run the identical procedure on
, we will get the same result. The axiom can be generalized for any predicate taking an arbitrary number of arguments.
Another axiom is required for lawless sequences. The ''axiom of density'', given by:
states that, for any finite prefix (encoded by)
, there is some sequence
beginning with that prefix. 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.
See also
*
Notes
References
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{{Refend
Constructivism (mathematics)