Editing Constructive analysis (section)

====Uncountability====
Recall that the preorder on [[cardinal number|cardinals]] "<math>\le</math>" in set theory is the primary notion defined as [[injective function|injection]] existence. As a result, the constructive theory of cardinal order can diverge substantially from the classical one. Here, sets like <math>{\mathbb Q}^{\mathbb N}</math> or some models of the reals can be taken to be [[Subcountability|subcountable]].

That said, [[Cantor's diagonal argument#In the absence of excluded middle|Cantors diagonal construction]] proving uncountability of powersets like <math>{\mathcal P}{\mathbb N}</math> and plain function spaces like <math>{\mathbb Q}^{\mathbb N}</math> is [[Intuitionistic logic|intuitionistically]] valid. Assuming <math>{\mathrm {PEM}}</math> or alternatively the [[countable choice]] axiom, models of <math>{\mathbb R}</math> are always uncountable also over a constructive framework.<ref>Bauer, A., Hanson, J. A. "The countable reals", 2022</ref> One variant of the diagonal construction relevant for the present context may be formulated as follows, proven using countable choice and for reals as sequences of rationals:<ref>See, e.g., Theorem 1 in Bishop, 1967, p. 25</ref>
:For any two pair of reals <math>a < b</math> and any sequence of reals <math>x_n</math>, there exists a real <math>z</math> with <math> a < z < b </math> and <math> \forall (n \in {\mathbb N}). x_n\, \#\, z</math>.
Formulations of the reals aided by explicit moduli permit separate treatments.

According to [[Akihiro Kanamori|Kanamori]], "a historical misrepresentation has been perpetuated that associates diagonalization with non-constructivity" and a constructive component of the [[Diagonal argument (proof technique)|diagonal argument]] already appeared in Cantor's work.<ref>[[Akihiro Kanamori]], "The Mathematical Development of Set Theory from Cantor to Cohen", ''[[Bulletin of Symbolic Logic]]'' / Volume 2 / Issue 01 / March 1996, pp 1-71</ref>