Subcountability
In constructive mathematics, a collection  is subcountable if there exists a partial surjection from the natural numbers onto it. This may be expressed as

where  denotes that  is a surjective function from a  onto . The surjection is a member of  and here the subclass  of  is required to be a set. In other words, all elements of a subcountable collection  are functionally in the image of an indexing set of counting numbers  and thus the set  can be understood as being dominated by the countable set . Discussion

Nomenclature 
Note that nomenclature of countability and finiteness properties vary substantially - in part because many of them coincide when assuming excluded middle. To reiterate, the discussion here concerns the property defined in terms of surjections onto the set  being characterized. The language here is common in constructive set theory texts, but the name subcountable has otherwise also been given to properties in terms of injections out of the set being characterized. The set  in the definition can also be abstracted away, and in terms of the more general notion  may be called a subquotient of . Example 
Important cases are where the set in question is some subclass of a bigger class of functions as studied in computability theory. There cannot be a computable surjection  from  onto the set of total computable functions , as demonstrated via the function  from the diagonal construction, which could never be in such a surjections image. However, via the codes of all possible partial computable functions, which also allows non-terminating programs, such subsets of functions, such as the total functions, are seen to be subcountable sets: The total functions are the range of some strict subset  of the natural numbers. Being dominated by an uncomputable, and so constructively uncountable, set of numbers, the name subcountable thus conveys that the constructively uncountable set  is no bigger than . Note that no effective map between all counting numbers  and the unbounded and non-finite indexing set  is asserted here, merely the subset relation . Being total is famously not a decidable property. By Rice's theorem on index sets, most domains of indices are, in fact, not computable sets. The demonstration that  is subcountable also implies that it is classically (non-constructively) formally countable, but this does not reflect any effective countability. In other words, the fact that an algorithm listing all total functions in sequence cannot be coded up is not captured by classical axioms regarding set and function existence. We see that, depending on the axioms of a theory, subcountability may be more likely to be provable than countability. Relation to excluded middle 
In constructive logics and set theories tie the existence of a function between infinite (non-finite) sets to questions of decidability and possibly of effectivity. There, the subcountability property splits from countability and is thus not a redundant notion. The indexing set  of natural numbers may be posited to exist, e.g. as a subset via set theoretical axioms like the separation axiom schema. Then by definition of , 

But this set may then still fail to be detachable, in the sense that

may not be provable without assuming it as an axiom. One may fail to effectively count the subcountable set  if one fails to map the counting numbers  into the indexing set , for this reason. Being countable implies being subcountable. In the appropriate context with Markov's principle, the converse is equivalent to the law of excluded middle, i.e. that for all proposition  holds . In particular, constructively this converse direction does not generally hold. In classical mathematics 
Asserting all laws of classical logic, the disjunctive property of  discussed above indeed does hold for all sets. Then, for nonempty , the properties numerable (which here shall mean that  injects into ), countable ( has  as its range), subcountable (a subset of  surjects into ) and also not -productive (a countability property essentially defined in terms of subsets of ) are all equivalent and express that a set is finite or countably infinite. Non-classical assertions 
Without the law of excluded middle, it can be consistent to assert the subcountability of sets that classically (i.e. non-constructively) exceed the cardinality of the natural numbers. Note that in a constructive setting, a countability claim about the function space  out of the full set , as in , may be disproven. But subcountability  of an uncountable set  by a set  that is not effectively detachable from  may be permitted. As  is uncountable and classically in turn provably not subcountable, the classical framework with its large function space is incompatible with the constructive Church's thesis, an axiom of Russian constructivism. Subcountable and ω-productive are mutually exclusive 
A set  shall be called -productive if, whenever any of its subsets  is the range of some partial function on , there always exists an element  that remains in the complement of that range. If there exists any surjection onto some , then its corresponding compliment as described would equal the empty set , and so a subcountable set is never -productive. As defined above, the property of being -productive associates the range  of any partial function to a particular value  not in the functions range, . In this way, a set  being -productive speaks for how hard it is to generate all the elements of it: They cannot be generated from the naturals using a single function. The -productivity property constitutes an obstruction to subcountability. As this also implies uncountability, diagonal arguments often involve this notion, explicitly since the late seventies. One may establish the impossibility of computable enumerability of  by considering only the computably enumerable subsets  and one may require the set of all obstructing 's to be the image of a total recursive so called production function. In set theory, where partial functions are modeled as collection of pairs, the space  given as  exactly holds all partial functions on  that have, as their range, only subsets  of . For an -productive set  one finds

Read constructively, this associates any partial function  with an element  not in that functions range. This property emphasizes the incompatibility of an -productive set  with any surjective (possibly partial) function. Below this is applied in the study of subcountability assumptions. Set theories

Cantorian arguments on subsets of the naturals 
As reference theory we look at the constructive set theory CZF, which has Replacement, Bounded Separation, strong Infinity, is agnostic towards the existence of power sets, but includes the axiom that asserts that any function space  is set, given  are also sets. In this theory, it is moreover consistent to assert that every set is subcountable. The compatibility of various further axioms is discussed in this section by means of possible surjections on an infinite set of counting numbers . Here  shall denote a model of the standard natural numbers. Recall that for functions , by definition of total functionality, there exists a unique return value for all values  in the domain, 

and for a subcountable set, the surjection is still total on a subset of . Constructively, fewer such existential claims will be provable than classically. The situations discussed below—onto power classes versus onto function spaces—are different from one another: Opposed to general subclass defining predicates and their truth values (not necessarily provably just true and false), a function (which in programming terms is terminating) does makes accessible information about data for all its subdomains (subsets of the ). When as characteristic functions for their subsets, functions, through their return values, decide subset membership. As membership in a generally defined set is not necessarily decidable, the (total) functions  are not automatically in bijection with all the subsets of . So constructively, subsets are a more elaborate concept than characteristic functions. In fact, in the context of some non-classical axioms on top of CZF, even the power class of a singleton, e.g. the class  of all subsets of , is shown to be a proper class. Onto power classes
Below, the fact is used that the special case  of the negation introduction law implies that  is contradictory. For simplicitly of the argument, assume  is a set. Then consider a subset  and a function . Further, as in Cantor's theorem about power sets, define

where,

This is a subclass of  defined in dependency of  and it can also be written

It exists as subset via Separation. Now assuming there exists a number  with  implies the contradiction

So as a set, one finds  is -productive in that we can define an obstructing  for any given surjection. Also note that the existence of a surjection  would automatically make  into a set, via Replacement in CZF, and so this function existence is unconditionally impossible. We conclude that the subcountability axiom, asserting all sets are subcountable, is incompatible with  being a set, as implied e.g. by the power set axiom. Following the above prove makes it clear that we cannot map  onto just  either. Bounded separation indeed implies that no set  whatsoever maps onto . Relatedly, for any function , a similar analysis using the subset of its range  shows that  cannot be an injection. The situation is more complicated for function spaces. In classical ZFC without Powerset or any of its equivalents, it is also consistent that all subclasses of the reals which are sets are subcountable. In that context, this translates to the statement that all sets of real numbers are countable. Of course, that theory does not have the function space set . Onto function spaces
By definition of function spaces, the set  holds those subsets of the set  which are provably total and functional. Asserting the permitted subcountability of all sets turns, in particular,  into a subcountable set. So here we consider a surjective function  and the subset of  separated as

with the diagonalizing predicate defined as

which we can also phrase without the negations as

This set is classically provably a function in , designed to take the value  for particular inputs . And it can classically be used to prove that the existence of  as a surjection is actually contradictory. However, constructively, unless the proposition  in its definition is decidable so that the set actually defined a functional assignment, we cannot prove this set to be a member of the function space. And so we just cannot draw the classical conclusion. In this fashion, subcountability of  is permitted, and indeed models of the theory exist. Nevertheless, also in the case of CZF, the existence of a full surjection , with domain , is indeed contradictory. The decidable membership of  makes the set also not countable, i.e. uncountable. Beyond these observations, also note that for any non-zero number , the functions  in  involving the surjection  cannot be extended to all of  by a similar contradiction argument. This can be expressed as saying that there are then partial functions that cannot be extended to full functions in . Note that when given a , one cannot necessarily decide whether , and so one cannot even decide whether the value of a potential function extension on  is already determined for the previously characterized surjection . The subcountibility axiom, asserting all sets are subcountable, is incompatible with any new axiom making  countable, including LEM. Models 
The above analysis affects formal properties of codings of . Models for the non-classical extension of CZF theory by subcountability postulates have been constructed. Such non-constructive axioms can be seen as choice principles which, however, do not tend to increase the proof-theoretical strengths of the theories much. There are models of IZF in which all sets with apartness relations are subcountable. CZF has a model in, for example, the Martin-Löf type theory . In this constructive set theory with classically uncountable function spaces, it is indeed consistent to assert the Subcountability Axiom, saying that every set is subcountable. As discussed, the resulting theory is in contradiction to the axiom of power set and with the law of excluded middle. More stronger yet, some models of Kripke–Platek set theory, a theory without the function space postulate, even validate that all sets are countable. The notion of size 
Subcountability as judgement of small size shall not be conflated with the standard mathematical definition of cardinality relations as defined by Cantor, with smaller cardinality being defined in terms of injections and equality of cardinalities being defined in terms of bijections. Constructively, the preorder "" on the class of sets fails to be decidable and anti-symmetric. The function space  (and also ) in a moderately rich set theory is always found to be neither finite nor in bijection with , by Cantor's diagonal argument. This is what it means to be uncountable. But the argument that the cardinality of that set would thus in some sense exceed that of the natural numbers relies on a restriction to just the classical size conception and its induced ordering of sets by cardinality. As seen in the example of the function space considered in computability theory, not every infinite subset of  necessarily is in constructive bijection with , thus making room for a more refined distinction between uncountable sets in constructive contexts. Motivated by the above sections, the infinite set  may be considered "smaller" than the class . Related properties
A subcountable set has alternatively also been called subcountably indexed. The analogous notion exists in which "" in the definition is replaced by the existence of a set that is a subset of some finite set. This property is variously called subfinitely indexed. In category theory all these notions are subquotients. See also
 Cantor's diagonal argument
 Computable function
 Constructive set theory
 Schröder–Bernstein theorem
 Subquotient
 Total order

References 

Constructivism (mathematics)