Subcountability

In constructive mathematics, a collection ${\displaystyle X}$ is subcountable if there exists a partial surjection from the natural numbers onto it. This may be expressed as

${\displaystyle \exists (I\subseteq \omega ).\,\exists f.\,(f:I\twoheadrightarrow X).}$

In other words, all elements of a subcountable collection ${\displaystyle X}$ are functionally in the image of a indexing set of counting numbers ${\displaystyle I\subseteq \omega }$ and thus the set ${\displaystyle X}$ can be understood as being dominated by the countable set ${\displaystyle \omega }$.

Asserting all laws of classical logic, the properties countable, subcountable and also not ${\displaystyle \omega }$-productive (explained below) are all equivalent and express that a set is finite or countably infinite. In contrast, in more constructive logics, which condition the existence of a bijection between infinite (non-finite) sets ${\displaystyle I}$ and ${\displaystyle \omega }$ to questions of effectivity and decidability, subcountability is not a redundant property.

Not asserting the law of excluded middle, in constructive set theories it can then also be consistent to assert the subcountability of sets that classically (i.e. non-constructively) would exceed the cardinality of the natural numbers, such as ${\displaystyle {\mathbb {R} }}$.