Related papers: At the End of Infinity
Bolzano and Cantor were the first mathematicians to make significant attempts to measure the size (numerosity) of different infinite collections. They differed in their methodological approaches, with Cantor's prevailing. This led to the…
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…
G\"odel proved in the 1930s in his famous Incompleteness Theorems that not all statements in mathematics can be proven or disproven from the accepted ZFC axioms. A few years later he showed the celebrated result that Cantor's Continuum…
Alternative set theory was created by the Czech mathematician Petr Vop\v enka in 1979 as an alternative to Cantor's set theory. Vop\v enka criticised Cantor's approach for its loss of correspondence with the real world. Alternative set…
It is well known that the set of algebraic numbers (let us call it $A$) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using \G1-based…
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…
The embracing of actual infinity in mathematics leads naturally to the question of comparing the sizes of infinite collections. The basic dilemma is that the Cantor Principle (CP), according to which two sets have the same size if there is…
Several examples are used to illustrate how we deal cavalierly with infinities and unphysical systems in physics. Upon examining these examples in the context of infinities from Cantor's theory of transfinite numbers, the only known…
Four constructions result from a desire to create enhancements to Cantor's infinite real set cardinality. Each continues to keep Cantor's cardinality formulation in place while providing new comparisons of arbitrary infinite sets. To…
We present a coherent collection of finite mathematical theorems some of which can only be proved by going well beyond the usual axioms for mathematics. The proofs of these theorems illustrate in clear terms how one uses the well studied…
We introduce a generalization of the Cantor-Dedekind continuum with explicit infinitesimals. These infinitesimals are used as numbers obeying the same basic rules as the other elements of the generalized continuum, in accordance with…
This text gives a rough, but linear summary covering some key definitions, notations, and propositions from Lambda Calculus: Its Syntax and Semantics, the classical monograph by Barendregt. First, we define a theory of untyped extensional…
The discovery of the infinite integer leads to a partition between finite and infinite numbers. Construction of an infinitesimal and infinitary number system, the Gossamer numbers. Du Bois-Reymond's much-greater-than relations and…
This paper is an investigation into Cantor works about representing a function with trigonometric series, and his proofs about its uniqueness. These works are important, because they cause invention of point-set topology, and foundation of…
We apply an inductive argument to three theorems of Cantor on (1) the uncountability of infinite binary sequences, (2) the uncountability of real numbers, and (3) the non-equinumerosity of sets with their powersets. This technique proves…
This paper examines the possibilities of extending Cantor's two arguments on the uncountable nature of the set of real numbers to one of its proper denumerable subsets: the set of rational numbers. The paper proves that, unless certain…
The mathematical study of infinity seems to have the ability to transport the mind to lofty and unusual realms. Decades ago, I was transported in this way by Rudy Rucker's book Infinity and the Mind. Despite much subsequent learning and…
For alternate Cantor real base numeration systems we generalize the result of Frougny and~Solomyak on~arithmetics on the set of numbers with finite expansion. We provide a class of alternate bases which satisfy the so-called finiteness…
Alan Turing is considered as a founder of current computer science together with Kurt Godel, Alonzo Church and John von Neumann. In this paper multiple new research results are presented. It is demonstrated that there would not be Alan…
The infinitary lambda calculi pioneered by Kennaway et al. extend the basic lambda calculus by metric completion to infinite terms and reductions. Depending on the chosen metric, the resulting infinitary calculi exhibit different notions of…