There is a model, for a system of intuitionistic analysis including Brouwer's principle for numbers and Kripke's schema, in which math formula ø-definable discrete sets of choice sequences are subcountable.

Based on the paper [4] we show that true second‐order arithmetic is interpretable over the real‐algebraic structure of models of intuitionistic analysis built upon a certain class of complete Heyting algebras.

We show that true first-order arithmetic is interpretable over the real-algebraic structure of models of intuitionistic analysis built upon a certain class of complete Heyting algebras. From this the undecidability of the structures follows. We also show that Scott's model is equivalent to true second-order arithmetic. In the appendix we argue that undecidability on the language of ordered rings follows from intuitionistically plausible properties of the real numbers.

Let $L = \langle, +, h_q, 1\rangle_{q \in \mathbb{Q}}$ where $\mathbb{Q}$ is the set of rational numbers and $h_q$ is a one-place function symbol corresponding to multiplication by $q$. Then the $L$-theory of Scott's model for intuitionistic analysis is decidable.