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Encyclopedia > Principia Mathematica

The Principia only covered set theory, cardinal numbers, ordinal numbers and real numbers; deeper theorems from real analysis were not included, but by the end of the third volume it was clear that all known mathematics could in principle be developed in the adopted formalism. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Alternative meaning: number of pitch classes in a set. ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... Please refer to Real vs. ... Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ...

The questions remained whether a contradiction could be derived from the Principia's axioms, and whether there exists a mathematical statement which could neither be proven nor disproven in the system. These questions were settled by Gödel's incompleteness theorem in 1931. Gödel's second incompleteness theorem shows that basic arithmetic cannot be used to prove its own consistency, so it certainly cannot be used to prove the consistency of anything stronger. In other words, the statement "there are no contradictions in the Principia system" cannot be proven true or false in the Principia system unless there are contradictions in the system (in which case it can be proven both true and false). Jump to: navigation, search In mathematical logic, GÃ¶dels incompleteness theorems are two celebrated theorems proven by Kurt GÃ¶del in 1931. ... Jump to: navigation, search 1931 is a common year starting on Thursday. ...

Yet, as Douglas Hofstadter [[1]] has pointed out, there may be additional levels of potential contradiction here. A central principle of the "system of types" mentioned above is that statements that are self referential are forbidden, to avoid Russell's paradox. Loops of statements that are self referential (circular definitions) are also forbidden. However, the statement "We do not allow self-referential statements in Principia Mathematica" is a violation of the rule against self-referential statements, an apparant contradiction at the heart of the philosophy. Russells paradox (also known as Russells antinomy) is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Cantor and Frege is contradictory. ...

A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustion upon completion of the third. Jump to: navigation, search Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ...

The Principia is widely considered by specialists in the subject to be one of the most important and seminal works in mathematical logic and philosophy.

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• "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." – page 362.

Begriffsschrift is the name of a book on logic by Gottlob Frege published in 1879. ...

Results from FactBites:

 Philosophiae Naturalis Principia Mathematica - Wikipedia, the free encyclopedia (2653 words) The Philosophiae Naturalis Principia Mathematica (Latin: "mathematical principles of natural philosophy", often Principia or Principia Mathematica for short) is a three-volume work by Isaac Newton published on July 5, 1687. Descartes' book of 1644 Principia philosophiae (Principles of philosophy) stated that bodies can act on each other only through contact: a principle that induced people, among them he himself, to conjecture an universal medium as the carrier of interactions such as light and gravity— the aether. It was perhaps the force of the Principia, which explained so many different things about the natural world with such economy, that caused this method to become synonymous with physics, even as it is practised almost three and a half centuries after his beginning.
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