quality:corruption
In the absence of excluded middle

Also in constructive mathematics, there is no surjection from the full domain N {\displaystyle {\mathbb {N} }} onto the space of functions N N {\displaystyle {\mathbb {N} }^{\mathbb {N} }} or onto the collection of subsets P (N) {\displaystyle {\mathcal {P}}({\mathbb {N} })}, which is to say these two collections are uncountable. Again using " < {\displaystyle <} " for proven injection existence in conjunction with bijection absence, one has N < 2 N {\displaystyle {\mathbb {N} }<2^{\mathbb {N} }} and S < P (S) {\displaystyle S<{\mathcal {P}}(S)}. Further, ¬ (P (S) ≤ S) {\displaystyle \neg ({\mathcal {P}}(S)\leq S)}, as previously noted. Likewise, 2 N ≤ N N {\displaystyle 2^{\mathbb {N} }\leq {\mathbb {N} }^{\mathbb {N} }}, 2 S ≤ P (S) {\displaystyle 2^{S}\leq {\mathcal {P}}(S)} and of course S ≤ S {\displaystyle S\leq S}, also in constructive set theory.

It is however harder or impossible to order ordinals and also cardinals, constructively. For example, the Schröder–Bernstein theorem requires the law of excluded middle. In fact, the standard ordering on the reals, extending the ordering of the rational numbers, is not necessarily decidable either. Neither are most properties of interesting classes of functions decidable, by Rice's theorem, i.e. the set of counting numbers for the subcountable sets may not be recursive and can thus fail to be countable. The elaborate collection of subsets of a set is constructively not exchangeable with the collection of its characteristic functions. In an otherwise constructive context (in which the law of excluded middle is not taken as axiom), it is consistent to adopt non-classical axioms that contradict consequences of the law of excluded middle. Uncountable sets such as 2 N {\displaystyle 2^{\mathbb {N} }} or N N {\displaystyle {\mathbb {N} }^{\mathbb {N} }} may be asserted to be subcountable. This is a notion of size that is redundant in the classical context, but otherwise need not imply countability. The existence of injections from the uncountable 2 N {\displaystyle 2^{\mathbb {N} }} or N N {\displaystyle {\mathbb {N} }^{\mathbb {N} }} into N {\displaystyle {\mathbb {N} }} is here possible as well. So the cardinal relation fails to be antisymmetric. Consequently, also in the presence of function space sets that are even classically uncountable, intuitionists do not accept this relation to constitute a hierarchy of transfinite sizes. When the axiom of powerset is not adopted, in a constructive framework even the subcountability of all sets is then consistent. That all said, in common set theories, the non-existence of a set of all sets also already follows from Predicative Separation.

In a set theory, theories of mathematics are modeled. Weaker logical axioms mean fewer constraints and so allow for a richer class of models. A set may be identified as a model of the field of real numbers when it fulfills some axioms of real numbers or a constructive rephrasing thereof. Various models have been studied, such as the Cauchy reals or the Dedekind reals, among others. The former relate to quotients of sequences while the later are well-behaved cuts taken from a powerset, if they exist. In the presence of excluded middle, those are all isomorphic and uncountable. Otherwise, variants of the Dedekind reals can be countable or inject into the naturals, but not jointly. When assuming countable choice, constructive Cauchy reals even without an explicit modulus of convergence are then Cauchy-complete and Dedekind reals simplify so as to become isomorphic to them. Indeed, here choice also aids diagonal constructions and when assuming it, Cauchy-complete models of the reals are uncountable.