What is the difference between a lawlike sequence of only zeros and a lawless sequence of only zeros in Intuitionism In our class the profosser mentioned that there is a difference between a lawlike sequence of only zeros and a lawless sequence of only zeros in Intuitionism but I didn't quiet understood.
In both we can think of it as a cauchy sequence and the real number is the limit which is zero, am I wrong ?
 A: When talking about lawless sequences, it's important to keep in mind what the a priori possible terms of the sequence are; usually they are either natural numbers, or sometimes just $0$ and $1$.  In the degenerate case where the only a priori possibility is $0$, there would be no difference between a lawless sequence and a lawlike one. I confess that I've never seen this degenerate case  mentioned, and the only reason I mention it here is to avoid its coming up as an exception to what I write below.  So from now on, I'll assume that there is at least one nonzero a priori possibility, say $1$.
In this situation (which is undoubtedly what the professor had in mind) I don't think "a lawless sequence of only zeros" makes sense from an intuitionistic point of view.  According to intuitionism, all one can ever know about a lawless sequence is that it is lawless, that the a priori possible terms are such-and-such, and that a finite initial segment of the sequence is so-and-so. Even if all of the finitely many terms already chosen are $0$ we can't know that all the future choices will also be $0$.  To talk about a lawless sequence consisting of only $0$'s, we'd have to adopt the viewpoint of the whole sequence (all of its terms) already having been chosen, and (as far as I know) this viewpoint is not admitted in intuitionistic mathematics.
A: See :


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*Anne Troelstra & Dirk van Dalen, Constructivism in mathematics: An Introduction. Volume I (1988), page 207:


Both are varieties of choice sequence, a peculiar object of Intuitionistic mathematics.
A choice sequnce is a sort of process not necessarily predetermined by a law or algorithm.
A lawlike sequence $a$ is a sequence $\langle a_n \rangle$ of numbers generated by a "law": a procedure, an algorithm.
From a non-Brouwerian point of view : "lawlike" = "recursive".
A lawless sequence $\alpha$ of e.g. natural is a process of choosing its values $\alpha_0, \alpha_1, \ldots \in \mathbb N$ such that at each stage in the generation of $\alpha$ we know only finitely many values.
At each stage $n$ of the process we are not able to impose general restrictions on future values; we cannot, for example, decide that all values to be chosen after stage $n$ are to be even.
Thus, we can never assert that its value will coincide with the values of some lawlike sequence.
Consider the variable $a$ ranging over lawlike sequences; in order to assert $\exists a \ (\alpha = a)$, we must know at some stage that $\alpha=a$ for some lawlike $a$, which is not. Thus, we have:

$\forall \alpha \ \lnot \exists a \ (\alpha=a)$.


See :


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*Errett Bishop & Douglas Bridges, Constructive Analysis (1985), page 9 for criticism:



[Brouwer] introduced the method of free-choice sequences for constructing the continuum, as a consequnce of which the continuum canno be discrete because it is not well enough defined.


See also Intuitionism : The continuum on choice sequences, both lawlike and lawless.
