Subcountability Explained
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
f\colonI\twoheadrightarrowX
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 those where the set in question is some subclass of a bigger class of functions as studied in computability theory. For context, recall that being total is famously not a decidable property of functions. Indeed, Rice's theorem on index sets, most domains of indices are, in fact, not computable sets.
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 set of natural numbers, the name
subcountable thus conveys that the set
is no bigger than
. At the same time, for some particular restrictive constructive semantics of function spaces, in cases when
is provenly not
computably enumerable, such
is then also not
countable, and the same holds for
.
Note that no effective map between all counting numbers
and the unbounded and non-finite indexing set
is asserted in the definition of subcountability - merely the subset relation
. A demonstration that
is subcountable at the same time 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
{N}\twoheadrightarrow{N}N
, may be disproven. But subcountability
of an uncountable set
by a set
that is not effectively detachable from
may be permitted.
A constructive proof is also classically valid. If a set is proven uncountable constructively, then in a classical context is it provably not subcountable. As this applies to
, 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.
[1] 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.
denotes the space that exactly hold all the partial functions on
that have, as their range, only subsets
of
.In set theory, functions are modeled as collection of pairs. Whenever
is a set, the set of sets of pairs
} X^I may be used to characterize the space of partial functions on
. The for an
-productive set
one finds
\forall(w\in({N}\rightharpoonupX)).\exists(d\inX).\forall(n\in{N}).w(n) ≠ d.
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
f\colonI\twoheadrightarrow{lP}{N}
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
\{y\inY\mid\exists(S\in{lP}Y).y=h(S)\landy\notinS\}
shows that
cannot be an injection. The situation is more complicated for function spaces.
[2] 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
f\colonI\twoheadrightarrow{N}N
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
{N}\twoheadrightarrow{N}N
, 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.
}. 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
References
- Gert Smolka, Skolems paradox and constructivism, Lecture Notes, Saarland University, Jan. 2015
- Bauer, A. "An injection from N^N to N", 2011