Related papers: Uncountable sets and an infinite real number game
For any particularly interesting theorem one proof is never enough. Instead, the first proof sets the challenge to find a more elegant method that illuminates subtle features of the math, is simpler to understand, or even avoids using…
A proof that the set of real numbers is denumerable is given.
Cantor's famous proof of the non-denumerability of real numbers does apply to any infinite set. The set of exclusively all natural numbers does not exist. This shows that the concept of countability is not well defined. There remains no…
An infinite game on the set of real numbers appeared in Matthew Baker's work [Math. Mag. 80 (2007), no. 5, pp. 377--380] in which he asks whether it can help characterize countable subsets of the reals. This question is in a similar spirit…
The uncountability of the reals was first established by Cantor in what was later heralded as the first paper on set theory. Since the latter constitutes the official foundations of mathematics, the logical study of the uncountability of…
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.
Matthew Baker investigated, in previous work, an elegant, infinite-length game that may be used to study subsets of real numbers. We present two accessible examples of how an important technique from set theory, or a different technique…
In this paper we introduce the concept of completeness of sets. We study this property on the set of integers. We examine how this property is preserved as we carry out various operations compatible with sets. We also introduce the problem…
We will show that, consistently, every uncountable set can be continuously mapped onto a non measure zero set, while there exists an uncountable set whose all continuous images into a Polish space are meager.
A Q-set is an uncountable set of reals all of whose subsets are relative $G_\delta$ sets. We prove that, for an arbitrary uncountable cardinal kappa, there is consistently a Q-set of size $\kappa$ whose square is not Q. This answers a…
We provide game-theoretic proofs of some well-known existence theorems of Friedberg numberings for the class of all partial computable functions, including (1) the existence of two incomparable Friedberg numberings; (2) the existence of a…
In this note we give an example of a nonmeasurable set in the probability space for an infinite sequence of coin flips. The example arises naturally from the notion of an equivariant function, and serves as a pedagogical illustration of the…
Continuity of measure asserts that the measure of the union of an increasing sequence of sets is equal to the supremum of the measures of those sets. We provide counter examples in the case of uncountable unions. We construct the first…
In Combinatorial Game Theory, short game forms are defined recursively over all the positions the two players are allowed to move to. A form is decomposable if it can be expressed as a disjunctive sum of two forms with smaller birthday. If…
The language of finite games is used to rephrase Pelant's proof of his result: The separable modification of the complete metric space $C([0,\omega_1])$ is not complete.
Given a coloring of the k-element subsets of an uncountable separable metric space, we show that there exists an infinite monochromatic subset which contains its limit point.
We prove that, for any $n\geq 0$, there exists an uncountable, $n$-dimensional, excellent, regular local ring with countable spectrum.
A set is introreducible if it can be computed by every infinite subset of itself. Such a set can be thought of as coding information very robustly. We investigate introreducible sets and related notions. Our two main results are that the…
In this paper I introduce a new and intuitive first-order foundational theory (where the concept of set is not primitive) and use it to show that the power set of an infinite set does not exist. In particular, proofs of uncountability of a…
We characterize countable dimensionality and strong countable dimensionality by means of an infinite game.