Matemática Discreta
325 views | +0 today
Follow
Your new post is loading...
Your new post is loading...
Scooped by André Souza Lemos
Scoop.it!

Russell's paradox - Wikipedia, the free encyclopedia

In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction. The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl and other members of the University of Göttingen.

According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell's paradox. Symbolically:

In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zermelo set theory, the first constructed axiomatic set theory. Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo–Fraenkel set theory (ZF).[1]

more...
No comment yet.
Scooped by André Souza Lemos
Scoop.it!

Khan Academy

Khan Academy | Matemática Discreta | Scoop.it
With a library of over 3,000 videos covering everything from arithmetic to physics, finance, and history and hundreds of skills to practice, we're on a mission to help you learn what you want, when you want, at your own pace.
André Souza Lemos's insight:

Para estudar (ou relembrar) a matemática fundamental (e até alguns tópicos de matemática superior).

more...
No comment yet.
Scooped by André Souza Lemos
Scoop.it!

What do grad students in math do all day?Edit Do they just sit at their desk an - Pastebin.com

more...
No comment yet.
Scooped by André Souza Lemos
Scoop.it!

Non-well-founded set theory - Wikipedia, the free encyclopedia

Non-well-founded set theories are variants of axiomatic set theory that allow sets to contain themselves and otherwise violate the rule of well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axioms implying its negation.

The study of non-well-founded sets was initiated by Dmitry Mirimanoff in a series of papers between 1917 and 1920, in which he formulated the distinction between well-founded and non-well-founded sets; he did not regard well-foundedness as an axiom. Although a number of axiomatic systems of non-well-founded sets were proposed afterwards, they did not find much in the way of applications until Peter Aczel's hyperset theory in 1988.[1]

The theory of non-well-founded sets has been applied in the logical modelling of non-terminating computational processes in computer science (process algebra and final semantics), linguistics and natural language semantics (situation theory), philosophy (work on the Liar Paradox), and in a different setting, non-standard analysis.[2]>

André Souza Lemos's insight:

Há conjuntos que contêm a si mesmos? Chamam-se conjuntos não-bem-fundamentados.

more...
No comment yet.
Scooped by André Souza Lemos
Scoop.it!

Connexive logic - Wikipedia, the free encyclopedia

Connexive logic names one class of alternative, or non-classical, logics designed to exclude the so-called paradoxes of material implication. (Other logical theories with the same agenda include relevance logic, also known as relevant logic.) The characteristic that separates connexive logic from other non-classical logics is its acceptance of Aristotle's Thesis, i.e. the formula,

as a logical truth. Aristotle's Thesis asserts that no statement follows from its own denial. Stronger connexive logics also accept Boethius' Thesis,

which states that if a statement implies one thing, it does not imply its opposite.

more...
No comment yet.
Scooped by André Souza Lemos
Scoop.it!

Charles Babbage and his Difference Engine #2

[Recorded: April 2008] Charles Babbage (1791-1871), computer pioneer, designed the first automatic computing engines. He invented computers but failed to bui...
more...
No comment yet.