Related papers: On Cantor's important proofs
We study two relations on multi-dimensional subshifts: A pre-order based on the patterns configurations contain and the Cantor-Bendixson rank. We exhibit several structural properties of two-dimensional subshifts: We characterize the…
Using an iterative tree construction we show that for simple computable subsets of the Cantor space Hausdorff, constructive and computable dimensions might be incomputable.
We introduce a proof-theoretic approach to showing nondefinability of second-order intuitionistic connectives by quantifier-free schemata. We apply the method to prove that Taranovsky's "realizability disjunction" connective does not admit…
We discuss some examples that illustrate the countability of the positive rational numbers and related sets. Techniques include radix representations, Godel numbering, the fundamental theorem of arithmetic, continued fractions, Egyptian…
Contrary to popular misconception, the question in the title is far from simple. It involves sets of numbers on the first level, sets of sets of numbers on the second level, and so on, endlessly. The infinite hierarchy of the levels…
Below, by space we mean a separable metrizable zero-dimensional space. It is studied when the space can be embedded in a Cantor set while maintaining the algebraic structure. Main results of the work: every space is an open retract of a…
We determine the proof-theoretic strength of the principle of countable saturation in the context of the systems for nonstandard arithmetic introduced in our earlier work.
Cantor's ternary function is generalized to arbitrary base-change functions in non-integer bases. Some of them share the curious properties of Cantor's function, while others behave quite differently.
Let $p/q$ ($p, q \in \mathbb{N}^*$) be a positive rational number such that $p > q^2$. We show that for any $\epsilon > 0$, there exists a set $A(\epsilon) \subset [0, 1[$, with finite border and with Lebesgue measure $< \epsilon$, for…
When working in NF, [1] there is a sense that there are more non-Cantorian sets than Cantorian sets. But it is not that immediate result as one expects, since they are externally equinumerous, and the qualification "Cantorian" is not…
After highlighting the cases in which the semantics of a language cannot be mechanically reproduced (in which case it is called inherent), the main epistemological consequences of the first incompleteness Theorem for the two fundamental…
We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete 1-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $M^{eq}$ has…
We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are (commutative unital) reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with…
These lecture notes cover classical undecidability results in number theory, Hilbert's 10th problem and recent developments around it, also for rings other than the integers. It also contains a sketch of the authors result that the integers…
The paper introduces the notion of the size of countable sets that preserves the Part-Whole Principle and generalizes the notion of the cardinality of finite sets. The sizes of natural numbers, integers, rational numbers, and all their…
In the absence of the axiom of choice, new results concerning sequential, Fr\'echet-Urysohn, $k$-spaces, very $k$-spaces, Loeb and Cantor completely metrizable spaces are shown. New choice principles are introduced. Among many other…
We first show that in the function realizability topos every metric space is separable, and every object with decidable equality is countable. More generally, working with synthetic topology, every $T_0$-space is separable and every…
We show that separability and second-countability are first-order properties among topological spaces definable in o-minimal expansions of $(\mathbb{R},<)$. We do so by introducing first-order characterizations -- definable separability and…
In 1984, Kurt Mahler posed the following fundamental question: How well can irrationals in the Cantor set be approximated by rationals in the Cantor set? Towards development of such a theory, we prove a Dirichlet-type theorem for this…
A dependent theory is a (first order complete theory) T which does not have the independence property. A main result here is: if we expand a model of T by the traces on it of sets definable in a bigger model then we preserve its being…