natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
Let $S$ be any set, and let $\mathbf{N}$ be the set of natural numbers. Let $|S|$ be the cardinality of $S$, and let $\aleph_0$ be the cardinality of $\mathbf{N}$.
Then $S$ is:
Note that the first two terms are not entirely standardised; some authors use them interchangeably for one or the other concept.
If you accept the axiom of choice, then there is really nothing more to say than what was above. In weaker foundations, more care may be needed. The following definitions should be more than enough for practical use in constructive mathematics:
Of course, the terms are even less standardised here. Certainly it is common for ‘countable’ to mean countably indexed, and yet it may also (as in classical mathematics) mean denumerable. In at least some contexts, ‘enumerable’ (without the initial ‘d’) means denumerably indexed.
Not only does the difference between ‘denumerable’ and ‘countable’ disappear when we add the prefix ‘sub’ (for obvious reasons), but it almost disappears when we add the suffix ‘ly indexed’! Specifically, a set is denumerably indexed if and only if it is both inhabited and countably indexed; and a set is countably indexed if and only if it is either empty or denumerably indexed. That countability involves lower subsets of $\mathbb{N}$ is also almost irrelevant here; any inhabited detachable subset of $\mathbb{N}$ is denumerably indexed regardless of its closure under order.
On the other hand, it is key that countability involves detachable subsets of $\mathbb{N}$, so that not every subcountable set is necessarily countable (or even countably indexed). Indeed, the set of computable numbers is subcountably indexed, while the set of real numbers is uncountable (and thus not countably indexed); yet in Russian constructivism, these are the same set! One reason to focus on countably indexed sets is that this is about the most general concept that definitely does not hold of the set of real numbers.
In addition, in the presence of countable choice, the difference between denumerably indexed and split denumerably indexed, countably indexed and split countably indexed, and subcountably indexed and split subcountably indexed disappear, since every surjection out of $\mathbb{N}$ and every subset of $\mathbb{N}$ has a section by countable choice.
Under the BHK interpretation of constructive mathematics, one only has the split versions of denumerably indexed, countably indexed, subcountably indexed.
There are a number of different definitions of an uncountable set, which all differ in the absence of countable choice:
Similarly as the case for above, the terms are even less standardised here. Certainly it is common for ‘uncountable’ to mean uncountably indexed, and yet it may also (as in classical mathematics) mean indenumerable.
We make the distinction between uncountable and uncountably indexed and between indenumerable and indenumerably indexed because in neutral constructive mathematics it is consistent that there exist a set which is both indenumerable and indenumerably indexed. However, similarly to countably indexed sets, in the presence of countable choice, the difference between uncountably indexed and split uncountably indexed and indenumerably indexed and split indenumerably indexed disappear.
There is a lot of freedom in defining uncountably indexed sets. You can start with countable and require that any function from them to $S$ be non-surjective (in the strong sense used in the definition above). Or you can start with denumerable or denumerably indexed sets and require that any function from them to $S$ be non-surjective, then add the additional clause that $S$ must be inhabited. Or you can compromise, and start with a detachable subset of $\mathbb{N}$ without requiring it to be a lower subset, since this is enough to prove $S$ inhabited and make the order-closure irrelevant. All of these definitions are equivalent!
In classical mathematics, the set of real numbers is uncountable. Even in constructive mathematics, the MacNeille real numbers, the Dedekind real numbers, and the Cauchy real numbers are all uncountable, in the sense of having a bijection with a countable set.
More generally, it is impossible for the real numbers to be both subcountable and countably indexed at the same time; subcountability implies that the real numbers have decidable equality because the natural numbers have decidable equality and injections preserve and reflect decidable equality, from which one can derive WLPO allowing one to prove that the real numbers are uncountably indexed by any of the usual arguments, resulting in a contradiction.
In fact, WLPO itself is sufficient to prove that the real numbers are uncountably indexed. However, it is consistent that, absent WLPO, the Dedekind real numbers are denumerably indexed and thus not uncountably indexed (cf Bauer 2022).
Every denumerable set is infinite, and hence so is every set with a denumerable subset. Classically (usng dependent choice $DC$), we have the converse: a set is infinite if and only if it has a denumerable subset. If a set has a denumerable subset, then its power set is uncountable. Even without $DC$, using excluded middle instead, the power set of an infinite set is uncountable.
In contrast, the set of algebraic numbers is denumerable. Also, the set of computable numbers is subcountably indexed (and infinite, so denumerable in classical mathematics). Combined with the uncountability of the real numbers, this gives the existence of transcendental numbers (constructively) and uncomputable numbers (classically).
Classically, every set is countably indexed xor uncountably indexed. Even constructively, a countably indexed set cannot be uncountably indexed. In Russian constructivism, however, there are subcountably indexed sets that are also uncountably indexed. In particular, the set of computable numbers is subcountably indexed, while the set of real numbers is uncountably indexed, yet in Russian constructivism, these are the same set! In some formalisms of Russian constructivism, every set is subcountably indexed.
The empty set is countable, although not denumerably indexed. Conversely, any uncountable set must be inhabited, as must be any denumerably indexed set. Even constructively, a countable set is empty xor inhabited.
Every finite set is countable, every finitely indexed set is countably indexed, every subfinite set is subcountable, and every subfinitely indexed set is subcountably indexed. Conversely, every uncountable set is infinite.
Classically, a denumerable set is precisely an infinite countable set; sometimes this is written as a countably infinite set. Thus classically, a countable set is finite xor denumerable. Even constructively, a countable set is denumerable if and only if it is not finite.
Classically (using countable choice), countable unions of countable sets are countable.
Arguably, set theory as such begins with the 1873 correspondence between Georg Cantor and Richard Dedekind on countability. Although they tacitly used excluded middle throughout, generally speaking, their results were constructive if taken to be about countably indexed sets. They worked out the countability of the algebraic numbers and the uncountability of the real numbers, including both constructive and nonconstructive proofs of the existence of transcendental numbers. (This was the same year that Charles Hermite published a proof that $\mathrm{e}$ is transcendental, although Joseph Liouville had shown how to artificially construct transcendental numbers as early as 1851.)
In subsequent correspondence with Cantor, Karl Weierstrass stressed the usefulness of the countability of the algebraic numbers but downplayed the notion of uncountability. Still, he urged Cantor to publish. Since Weierstrass and the finitist Leopold Kronecker were senior editors at Krelle's Journal, Cantor wrote his 1874 article carefully. He showed the denumerability of the algebraic numbers explicitly and titled the article as if that were the only subject. He then added a footnote in the proofreading stages, using an interval-based argument, showing how a transcendental number could be constructed from this denumeration.
Cantor's argument was essentially a constructive proof that the set of real numbers is uncountable, as it could apply to any denumerably indexed subset. He tacitly used LLPO, but it's straightforward to avoid this using binary meets and joins of real numbers, which are constructive. He also used a Dedekind cut to construct the final transcendental number, and it is this that Kronecker would have most likely objected to, since it is an irreducibly infinite procedure to construct a single number. However, the argument is acceptable to any non-finitist constructivist.
In the correspondence with Dedekind, it was Cantor who first considered and proved the uncountability of the real numbers but Dedekind who first considered and proved the countability of the algebraic numbers, which was the ostensible subject of Cantor's article. Furthermore, it was Dedekind who first wrote the interval-based proof that Cantor used in his footnote that was the real point of the article. However, Cantor gave Dedekind no credit. This strained their relationship for a few years afterward.
The diagonal argument also shows the uncountability of the real numbers (nonconstructively through Cantor's Theorem on the cardinality of power sets, constructively through a slight generalization), but Cantor did not publish this until 1891. Interpreted finitistically as an algorithm to construct, from a sequence of real numbers, an approximation of a hypothetical number not in the sequence, the diagonal argument is more efficient; pedagogically, it is easier to explain. However, it was not the first.
which discusses both the origin of the 1874 article and the questions of constructivity that arose in its aftermath.
That in the absence of countable choice it is consistent that the Dedekind real numbers are denumerably indexed:
Andrej Bauer, The countable reals (video).
Andrej Bauer, James Hanson, The Countable Reals (arXiv:2404.01256)
Last revised on April 17, 2024 at 18:00:24. See the history of this page for a list of all contributions to it.