Academic literature on the topic 'Infinito Cantor numeri transfiniti'

La presente tesi si occupa, da un punto di vista matematico e filosofico, dello studio dei numeri transfiniti introdotti da Georg Cantor. Vengono introdotti i concetti di numero cardinale ed ordinale, la loro aritmetica ed i principali risultati riguardo al concetto di insieme numerabile. Si discutono le nozioni di infinito potenziale ed attuale e quella di esistenza secondo la concezione di Cantor. Viene infine presentata l'induzione transfinita, una generalizzazione al caso transfinito del principio di induzione matematica.

Thesis advisor: Patrick Byrne
In the late 19th century, Georg Cantor opened up the mathematical field of set theory with his development of transfinite numbers. In his radical departure from previous notions of infinity espoused by both mathematicians and philosophers, Cantor created new notions of transcendence in order to clearly described infinities of different sizes. Leading the opposition against Cantor's theory was Leopold Kronecker, Cantor's former mentor and the leading contemporary German mathematician. In their lifelong dispute over the transfinite numbers emerge philosophical disagreements over mathematical existence, consistency, and freedom. This thesis presents a short summary of Cantor's controversial theories, describes Cantor and Kronecker's philosophical ideas, and attempts to state clearly their differences of opinion. In the end, the author hopes to present the shock caused by Cantor's work and an appreciation of the two very different philosophies of mathematics represented by Cantor and Kronecker
Thesis (BA) — Boston College, 2005
Submitted to: Boston College. College of Arts and Sciences
Discipline: Philosophy
Discipline: College Honors Program

CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
Subjacente à teoria dos números ordinais transfinitos de Cantor, há uma perspectiva finitista. Segundo tal perspectiva, Deus pode bem ordenar o infinito usando, para tanto, de procedimentos similares ao ato de contar, entendido como o ato de bem ordenar o finito. Desta maneira, um diálogo natural entre Cantor e Dedekind torna-se possível, dado que Dedekind foi o primeiro a tratar o ato de contar como sendo, em sua essência, uma forma de bem ordenar o mundo espáciotemporal pelos números naturais. Nesta tese, o conceito de número ordinal transfinito, de Cantor, é entendido como uma extensão do conceito dedekindiano de número natural.
Underlying Cantor s transfinite ordinal numbers theory, there is a finistic perspective. Accordingly that perspective, God can well order the infinite using, for that, similar procedures to the act of counting, understood as the act of well order the finite. That s why a natural dialog between Cantor and Dedekind becomes possible, since Dedekind was the first to consider the act of counting as being, in its essence, a way of well order the spatial-temporal world by natural numbers. In this thesis, the concept of Cantor´s transfinite ordinal number is understood as an extension of dedekindian concept of natural number.

‘Counting infinity’ returns to the mathematics of infinity, discussing Cantor’s remarkable theory of how to count infinite sets, and the discovery that there are different sizes of infinity. For example, the set of all integers is infinite, and the set of all real numbers (infinite decimals) is infinite, but these infinities are fundamentally different, and there are more real numbers than integers. The ‘numbers’ here are called transfinite cardinals. For comparison, another way to assign numbers to infinite sets is mentioned, by placing them in order, leading to transfinite ordinals. It ends by asking whether the old philosophical distinction between actual and potential infinity is still relevant to modern mathematics, and examining the meaning of mathematical existence.

This chapter locates the roots of the Marxist theory of revolutionary change in G.W.F. Hegel’s philosophy. In the well-known formula, cumulative changes in quantitative properties give rise to a qualitative leap into the future. However, the chapter argues that the idea rests on shaky ontological foundations. Through a close reading of the Science of Logic, it is shown that Hegel’s idea of leaps relies on excising irrational numbers. To make his dialectical transitions work, Hegel has to dialecticise the mathematical infinite and ignore scientific epistemological breaks from the classical period onwards. This compares unfavourably to Alain Badiou, who makes Georg Cantor’s breakthrough with transfinite set theory the lynchpin for his discontinuous philosophy of events. The final section argues that Hegel’s notion of quantity to quality leaps is also complicit with the reformism and technological determinism promoted by key thinkers of Second International Marxism.