The Countable Reals (2404.01256v2)
Abstract: We construct a topos in which the Dedekind reals are countable. The topos arises from a new kind of realizability, which we call parameterized realizability, based on partial combinatory algebras whose application depends on a parameter. Realizers operate uniformly with respect to a given parameter set. Our construction uses a sequence of reals in $[0,1]$, discovered by Joseph Miller, that is non-diagonalizable in the sense that any real which is oracle-computable uniformly from representations of the sequence must already appear in it. When used as the parameter set, this yields a topos in which the non-diagonalizable sequence becomes an epimorphism onto the Dedekind reals, rendering them internally countable. The resulting topos is intuitionistic: it refutes both the law of excluded middle and countable choice. Nevertheless, much of analysis survives internally. The Cauchy reals are uncountable. The Hilbert cube is countable, so Brouwer's fixed-point theorem follows from Lawvere's. The intermediate value theorem and the analytic form of the lesser limited principle of omniscience hold, while the limited principle of omniscience fails. Although no real-valued map has a jump, it remains open whether all such maps are continuous. Finally, the closed interval $[0,1]$, being countable, can be covered by a sequence of open intervals of total length less than any $\epsilon > 0$, with no finite subcover. Yet, we show that any cover using intervals with rational endpoints must admit a finite subcover.
- Andrej Bauer. An injection from baire space to natural numbers. Mathematical Structures in Computer Science, 25(Special issue 7):1484–1489, November 2015.
- M. Beeson and A. Ščedrov. Church’s thesis, continuity, and set theory. Journal of Symbolic Logic, 49(2):630–643, 1984.
- Michael J. Beeson. Foundations of Constructive Mathematics, volume 3 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1985.
- Errett Bishop. Foundations of Constructive Analysis. McGraww-Hill, 1967.
- Ingo Blechschmidt. On the uncountability of the set of reals. Constructive news mailing list, June 2018. Available at https://groups.google.com/g/constructivenews/c/jSvzqu1LUis.
- A constructive Knaster-Tarski proof of the uncountability of the reals, 2019. arXiv:1902.07366.
- Compactness and continuity, constructively revisited. In Julian Bradfield, editor, Computer Science Logic, pages 89–102, Berlin, Heidelberg, 2002. Springer Berlin Heidelberg.
- Varieties of Constructive Mathematics. Number 97 in London Mathematical Society Lecture Note Series. Cambridge University Press, 1987.
- Georg Cantor. Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen. Journal für die Reine und Angewandte Mathematik, 1874(77):258–262, 1874.
- Georg Cantor. Ueber unendliche, lineare Punktmannichfaltigkeiten. Mathematische Annalen, 15:1–7, 1879.
- Fixed point theorems for multi-valued transformations. American Journal of Mathematics, 68(2):214, April 1946.
- Solomon Feferman. A language and axioms for explicit mathematics. In John Newsome Crossley, editor, Algebra and Logic, pages 87–139, Berlin, Heidelberg, 1975. Springer Berlin Heidelberg.
- Wilfrid Hodges. An editor recalls some hopeless papers. The Bulletin of Symbolic Logic, 4(1):1–16, 1998.
- J. M. E. Hyland. The effective topos. In A.S. Troelstra and D. Van Dalen, editors, The L.E.J. Brouwer Centenary Symposium, pages 165–216. North Holland Publishing Company, 1982.
- Tripos theory. Mathematical Proceedings of the Cambridge Philosophical Society, 88(2):205–232, September 1980.
- Phil Scott Jim Lambek. Introduction to Higher Order Categorical Logic. Cambridge University Press, 1986.
- Peter T. Johnstone. Sketches of an Elephant: A Topos Theory Compendium. Number 44 in Oxford logic guides. Oxford University Press, 2002.
- Stephen Cole Kleene. On the interpretation of intuitionistic number theory. Journal of Symbolic Logic, 10:109–124, 1945.
- Bronisław Knaster. Un théorème sur les fonctions d’ensembles. Annales Polonici Mathematici, 6:133–134, 1928.
- Partial recursive functionals and effective operations. In A. Heyting, editor, Constructivity in mathematics, pages 290–297. North-Holland, Amsterdam, 1959.
- F. W. Lawvere. Diagonal arguments and cartesian closed categories. Lecture Notes in Mathematics, 92:134–145, 1969. Republished in: Reprints in Theory and Applications of Categories, No. 15 (2006), 1–13.
- Sets for Mathematics. Cambridge University Press, 2003.
- Joseph S. Miller. Degrees of unsolvability of continuous functions. Journal of Symbolic Logic, 69(2):555–584, 2004.
- V. P. Orevkov. A constructive map of the square into itself, which moves every constructive point. Doklady Akademii Nauk SSSR, 152(2):55–58, 1963.
- Hartley Rogers. Theory of Recursive Functions and Effective Computability. McGraw-Hill Book Company, 1967.
- Thoralf Skolem. Einige Bemerkungen zu axiomatischen Begründung der Mengenlehre. In Matematikerkongressen i Helsingfors den 4-7 juli 1922; den femte Skandinaviska matematikerkongressen redogörelse, pages 217–232. Akademiska bokhandeln, Helsingfors, 1923.
- Robert I. Soare. Recursively Enumerable Sets and Degrees. Springer-Verlag, 1987.
- Alfred Tarski. A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics, 5(2):285–309, 1955.
- Paul Taylor. Practical Foundations of Mathematics. Number 59 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1999.
- Constructivism in Mathematics (volume 1), volume 121 of Studies in Logic and Foundations of Mathematics. North-Holland, 1988.
- G. S. Tseitin. Algorithmic operators in constructive metric spaces. Trudy Matematiki Instituta Steklov, 64(2):295–361, 1967. English translation: American Mathematical Society Translations, series 2, vol. 64, 1967, pp. 1-80.
- Jaap van Oosten. Realizability: An Introduction To Its Categorical Side, volume 152 of Studies in logic and the foundations of mathematics. Elsevier, 2008.
- The drinker paradox and its dual. Preprint arXiv:1805.06216, 2018.
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The big idea in simple terms
This paper builds a special mathematical “universe” (called a topos) where something surprising happens: the real numbers (understood as Dedekind reals) are countable. That means, inside this universe, you can list the real numbers one-by-one with the natural numbers, even though in ordinary math class (classical mathematics) we learn from Cantor’s famous argument that the reals are uncountable.
This doesn’t break math. It shows that whether the reals are countable can depend on the logical rules your universe obeys. The universe in the paper uses intuitionistic logic (which avoids some classical rules like the law of excluded middle and certain forms of the axiom of choice). In that setting, Cantor’s diagonal argument doesn’t go through in the usual way for the particular kind of reals considered.
Below is a friendly walkthrough of what the paper does and why it matters.
1) What is the paper about?
The paper creates a new kind of mathematical universe (a topos) where the Dedekind reals are countable. To do this, the authors introduce a new construction tool called a parameterized realizability topos. They then use a special kind of sequence of real numbers, invented by Joseph Miller, that “defeats” diagonalization in a uniform way. With this toolset, they show there is a map from the natural numbers onto the real numbers inside their universe, so the reals become countable there.
2) What are the main questions?
- Can we build a consistent universe of mathematics where the Dedekind reals are countable?
- What logical rules must such a universe follow (and which classical rules must it drop)?
- What special constructions are needed to stop Cantor’s diagonal argument from making the reals uncountable?
- What familiar theorems about real numbers still hold (or fail) in such a universe?
3) How do they do it? (Methods explained simply)
Think of a topos as a self-contained math world with its own rules. Inside, you interpret statements like “every real number has a certain property” in that world’s logic.
Two main ingredients power the construction:
1) Parameterized realizability (proofs as programs with a uniform helper): - Realizability is a way to interpret proofs as actual programs or procedures. A “realizer” is a bit like code that witnesses why a statement is true. - Here, realizers are allowed to consult an “oracle,” which you can picture as a helpful reference sheet or a black box they can query. But there’s a twist: the realizers must behave uniformly with respect to the oracle. Roughly, if different oracles represent the same underlying data (like the same sequence), the program must act the same way regardless of which encoding it gets. No cheating by depending on a specific encoding. - Technically, the authors define parameterized partial combinatory algebras (ppcas). These are abstract machines where application (running a program on an input) depends on a parameter (the oracle), but in a very controlled, uniform way. They show you can still do all the usual programming moves (booleans, pairs, numbers, recursion) uniformly.
2) Miller’s non-diagonalizable (or “Miller”) sequence: - Cantor’s diagonal trick normally says: given any list of real numbers, you can build a new real that differs from the nth listed real in the nth digit, so your list can’t be complete. - Miller found a special kind of sequence a(0), a(1), a(2), … in with the following property: any real number that can be computed from the sequence by an oracle program in a uniform way is already on the sequence. In short: “If you can compute it uniformly from the list, it’s already in the list.” - That blocks diagonalization in the intended sense, because the diagonal construction would have to be a uniform oracle program (it isn’t, under the uniformity requirement the authors enforce). - How do they know such a sequence exists? The paper outlines a proof strategy using nested intervals (to interpret partial computations as narrowing intervals) and a general fixed-point theorem (a strengthened form of Brouwer/Kakutani). Think of it as defining a process that, for each position n, says which values could possibly show up at slot n, given what’s allowed by all uniform computations. A fixed point of this process is a sequence that exactly matches its own “allowed outputs.” That fixed point is the Miller sequence.
Putting it together:
- The authors build a topos where realizers are programs that use oracles uniformly.
- They pick the parameter set to range over all oracles that represent one chosen Miller sequence.
- In that setting, the Miller sequence appears as a “surjective-like” map (an epimorphism) from the natural numbers onto the Dedekind reals, making them countable inside this topos.
4) What did they find, and why is it important?
Main results inside the new topos:
- The Dedekind reals are countable. There is an epimorphism (Dedekind reals).
- The topos is intuitionistic: it does not validate the law of excluded middle (“every statement is true or false”) and does not validate the axiom of countable choice.
- The Cauchy reals are still uncountable. So “real numbers” split into different notions with different sizes.
- The Hilbert cube is countable.
- Brouwer’s fixed-point theorem holds (in all finite dimensions), derived easily from Lawvere’s fixed-point theorem inside the topos. In dimension 1, this gives the Intermediate Value Theorem and a classical-looking principle called LLPO (the “lesser limited principle of omniscience”).
- Every function is continuous (a version of the Kreisel–Lacombe–Shoenfield–Tseitin theorem), because the standard proof works uniformly with oracles here.
- Compactness behaves strangely: you can cover by countably many open intervals whose total length is as small as you like (say less than any given ), and yet there is no finite subcover. However, if you restrict to open intervals with rational endpoints, any countable cover does have a finite subcover.
Why this matters:
- It shows that Cantor’s uncountability of the reals relies on classical logic and certain choice principles. If you change the logical backdrop and require uniformity in computation with oracles, diagonalization can be blocked in a precise way.
- It provides a clean new model (a topos) where analysts can explore which theorems of analysis survive, fail, or change when you use constructive/intuitionistic logic.
- The construction of parameterized realizability toposes is a new tool that can be adapted to other questions about computation, logic, and analysis.
5) What are the implications?
- Logic matters: The size and behavior of fundamental objects (like the real numbers) depend on the logical rules your math universe accepts. This clarifies the role of excluded middle and choice in classical proofs of uncountability.
- New playground for constructive analysis: The topos gives a consistent setting where some classical theorems still hold (fixed-point theorems, intermediate value), others get stronger (all real-valued maps are continuous), and some classical compactness facts fail in subtle ways. This helps map out the landscape of what is constructively valid.
- Techniques that travel: The idea of “uniform realizability with parameters” is broadly useful. It carefully separates what depends on the information content of a structure (like a sequence) from what depends on the way that information is encoded. That idea can inform future work in computability, logic, and category theory.
In short, the paper demonstrates that by carefully controlling what “computable from a list” means—specifically, insisting on uniform computation across all representations—you can build a rigorous universe where the Dedekind reals are countable. This reorients how we think about uncountability: not as an absolute fact about reals alone, but as a fact that also depends on the logical lens we choose.
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