A constructive Knaster-Tarski proof of the uncountability of the reals
aa r X i v : . [ m a t h . HO ] F e b A CONSTRUCTIVE KNASTER–TARSKI PROOF OF THEUNCOUNTABILITY OF THE REALS
INGO BLECHSCHMIDT AND MATTHIAS HUTZLER
Abstract.
We give an uncountability proof of the reals which relies on theirorder completeness instead of their sequential completeness. We use neither aform of the axiom of choice nor the law of excluded middle, therefore the proofapplies to the MacNeille reals in any flavor of constructive mathematics. Theproof leans heavily on Levy’s unusual proof of the uncountability of the reals.
One way to verify the uncountability of the reals is as follows. We first observethat the usual diagonalization technique shows that the powerset of the naturalsis uncountable. We then show that this powerset is in bijection with the reals. Inconstructive mathematics, more precisely the kind of mathematics which can becarried out in any topos or the kind of mathematics formalizable in IntuitionisticZermelo–Fraenkel set theory, the first step is still valid, while the second might fail.This failure may occur for any of the several possible flavors of the reals such asthe Cauchy reals, the Dedekind reals or the MacNeille reals (which all coincide inclassical mathematics, but might differ in constructive mathematics). Therefore adifferent approach is needed.This note gives a constructive proof that one of these flavors, the MacNeille re-als, is uncountable. To the best of our knowledge, this is the first result in thatdirection. However, it is just a baby step towards an understanding whether anyof the more interesting flavors of the reals can constructively be shown to be un-countable, a problem posed to us by Andrej Bauer who we gratefully acknowledge.The proof presented here is made possible because the MacNeille reals – unlike theCauchy or Dedekind reals – can constructively be shown to be (conditionally) ordercomplete [1, Lemma D4.7.7].Sensibilities of constructive mathematics aside, the proof presented here is inter-esting because it uses only the order completeness of the reals, not their sequentialcompleteness, and because it puts the Knaster–Tarski fixed point theorem to gooduse. This fixed point theorem is fundamental to theoretical computer science, butappears to be seldomly used in classical analysis. The proof is an adaptation ofLevy’s unusual proof [2].
Theorem.
Let f : N → R be a map. Then there is a number x ∈ R such that forno n ∈ N , f ( n ) = x .Proof. The map g : R −→ R , x sup M X n ∈ M − n , where M ranges over all those (Bishop-)finite subsets of N such that f [ M ] < x ,is well-defined (because the sets the suprema are taken of are inhabited by zeroand bounded from above by 2), monotone, has a postfixpoint (0 ≤ g (0)), and has EFERENCES 2 an upper bound for all of its postfixpoints (if x ≤ g ( x ), then x ≤ x .If x = f ( n ) for a number n ∈ N , then for any finite subset M of N suchthat f [ M ] < x , X n ∈ M − n + 2 − n = X n ∈ M ∪{ n } − n ≤ g ( x + 2 − n ) , hence g ( x + 2 − n ) ≥ g ( x ) + 2 − n ≥ x + 2 − n . Thus x + 2 − n is a greaterpostfixpoint than x , a contradiction. (cid:3) It is possible to unwind the application of the Knaster–Tarski fixed point theoremto obtain an entirely elementary proof of uncountability. This unwinding makes theimpredicative nature of the proof manifest.
Second proof.
We consider the same map g : R → R as in the first proof. Let x bethe supremum of the set A := { x ∈ R | x ≤ g ( x ) } ; this supremum exists because A is inhabited (by zero) and bounded from above (by 2).If x = f ( n ) for a number n ∈ N , then x − − n is an upper bound for A ,contradicting the fact that x is the least upper bound of A : Let x ∈ A . If x >x − − n , then g ( x + 2 − n ) ≥ g ( x ) + 2 − n ≥ x + 2 − n (where the first inequality isas in the first proof, exploiting that x ≤ x by definition of x ), hence x + 2 − n ∈ A ,thus x + 2 − n ≤ x , a contradiction. Hence x ≤ x − − n . (cid:3) References [1] P. T. Johnstone.
Sketches of an Elephant: A Topos Theory Compendium . Ox-ford University Press, 2002.[2] E. Levy. “An unusual proof that the reals are uncountable”. arXiv:0901.0446[math.HO]. 2009.
Universit`a di Verona, Department of Computer Science, Strada le Grazie 15, 37134Verona, Italy
E-mail address : [email protected] Universit¨at Augsburg, Institut f¨ur Mathematik, Universit¨atsstr. 14, 86159 Augs-burg, Germany
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