I have the distinct memory of having often heard and read that intuitionism was inter alia geared to avoid Cantor's uncountable sets, and it may be that this was Brouwer's plan. But are there accounts which demonstrate that early intuitionism (i.e. before the advent of modern intuitionistic set theories, which either do not have the power set or do accept Cantor's conclusion) had some intuitionistically reasonable way of evading Cantor's uncountable sets?



Did Brouwer evade uncountability?
You are probably referring to Brouwer's considerations of the Creative Subject, which can be formulated mathematically as *Kripke's schema*. It implies that all subsets of $\mathbb{N}$ are countable, for example. I am having trouble finding good references, maybe these two will get you started:


* Göran Sundholm: ["Constructive recursive functions, Church's thesis and Kripke's schema"](http://www.academia.edu/3528209/Constructive_recursive_functions_Church_thesis_and_Kripkes_schema)
* ["The Use of Kripke's Schema as a Reduction Principle"](http://www.jstor.org/stable/2272124) D. Van Dalen
The Journal of Symbolic Logic
Vol. 42, No. 2 (Jun., 1977), pp. 238-240


I know little about the history of Brouwer's mathematics, but I would be very much surprised to hear that he set out to demolish Cantor's set theory. I thought his criticism was pointed at Hilbert's purely existential proofs, not at Cantor. I also never heard that uncountability was considered a problem, it was rather methods of proof.


From a purely mathematical point of view (i.e., ignoring history) it makes no sense to "avoid uncountability" because the usual diagonalization proofs of uncountability of Baire space $\mathbb{N}^{\mathbb{N}}$, Cantor space $\{0,1\}^{\mathbb{N}}$ and powerset $\mathcal{P}(\mathbb{N})$ are intuitionistically valid, and Brouwer would have of course known that. It would be hard for Brouwer to avoid these spaces, especially the Baire and the Cantor spaces, as these correspond to the totalities of all paths through a spread and through a fan, respectively. At best some limited form of "everything is countable" is tenable, for instance "every subset of $\mathbb{N}$ is countable" – which is of course valid classically but not in pure intuitionistic logic.


