Related papers: There are only countably many sets

This somewhat unusual proof for the fact that the reals are uncountable, which is adapted from one of Bourbaki's proofs in "Fonctions d'une variable reelle", may be of some interest.

These notes provide an opportunity to discover the beauty of Bourbaki set theory, and I hope that they will facilitate the task to those who find it difficult to read this book, one of the most critical elements of the mathematics of…

We give a short proof of the well-known fact that the unit interval [0,1] is uncountable by means of a simple infinite game. We also show using this game that a (non-empty) perfect subset of [0,1] must be uncountable.

We disprove a 2002 conjecture of Dombi from additive number theory. More precisely, we find examples of sets $A \subset \mathbb{N}$ with the property that $\mathbb{N} \setminus A$ is infinite, but the sequence $n \rightarrow |\{ (a,b,c) \,…

A proof that the set of real numbers is denumerable is given.

It is proved that all recursively enumerable sets of natural numbers can be represented by arithmetic formulas (of two kinds) with only 3 quantifiers.

We isolate a normal-form mechanism underlying Bourbaki--Witt fixed-point arguments and least-upper-bound versions of Zorn-type maximality principles. Given a progressive self-map on a partially ordered set, we define a Bourbaki tower as a…

It is unprovable that every complete subalgebra of a countably closed complete Boolean algebra is countably closed.

There is a serious mistake in the proof.

We prove that finite sets of real numbers satisfying $|AA| \leq |A|^{1+\epsilon}$ with sufficiently small $\epsilon > 0$ cannot have small additive bases nor can they be written as a set of sums $B+C$ with $|B|, |C| \geq 2$. The result can…

We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.

We prove that the Fibonacci Lie algebra and the related just infinite self-similar Lie algebra are not finitely presented.

Assuming the abc conjecture with $\epsilon=1/6$, we use elementary methods to show that only finitely many $s$-Cullen numbers are repunits, aside from two known infinite families. More precisely, only finitely many positive integers $s$,…

For $c>0$, let $X_c$ denote the set of $x\in\mathbb{R}\backslash\mathbb{Q}$ such that $\left| x-\frac{p}{q} \right|<\frac{1}{cq^2}$ has only finitely many rational solutions $\frac{p}{q}$. It is a classical fact, known since the 1950s, that…

Our main result (Theorem A) shows the incompleteness of any consistent sequential theory T formulated in a finite language such that T is axiomatized by a collection of sentences of bounded quantifier-alternation-depth. Our proof employs an…

We describe geometrically and algebraically the set of unattainable points for the Rational Hermite Interpolation Problem (i.e. those points where the problem does not have a solution). We show that this set is a union of equidimensional…

Borsuk's conjecture states that any bounded set in R^n can be partitioned into n+1 sets of smaller diameter. It is known to be false for all n bigger or equal to 323. Here we show that Borsuk's conjecture fails in dimensions 321 and 322.…

We present two sets of 12 integers that have the same sets of 4-sums. The proof of the fact that a set of 12 numbers is uniquely determined by the set of its 4-sums published 50 years ago is wrong, and we demonstrate an incorrect…

Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every…

We prove Union-Closed sets conjecture.