Anyone studying something that has to do with mathematics in some way comes to the issue of quantities is a must. And then some professor and says one:
by Cantor (1877) is a set the amount referred to . whole " Then you think:" Oh great, then everything is all clear. " It is of course true that one has worked before 1877 even with quantities, and certainly had a similar idea. Only one had to do less interested in the basics, and most of these mathematicians even today reluctantly.
The formulation of a set theory was thus actually something new. Set theory, based on the Cantor set term means naive set theory. Why "naive", it is equally to be clear. a good professor also mentioned that this idea of a set has his hook, that is not consistent. Here mostly is the concept of Russell's antinomy
. Stating that the set of all sets that do not contain themselves as an element, even an amount that does not contain themselves. But because they do not even contain it contains itself This is obviously a paradox. Russell himself has formulated this paradox also popularly understood, in the form of the famous
barber paradox. Who wants this to know more, to http://de.wikipedia.org/wiki/Barbier-Paradoxon make smart, a very amusing article. So it is not so loose from the stool, as formulated by the Cantor set theory. At least not when in a proof with amounts such deals, that such paradoxes can occur. Do you need a more reliable basis. Thus it was necessary to put the set theory on solid ground. For this there were different approaches and accepts most likely is named after its inventors Zermelo-Fraenkel set theory
short
ZF that has found its fulfillment 1930th This has 9 axioms that have been using a so-called predicate logic the first stage formulated with identity . "what is a predicate logic the first stage of identity?"
one wonders. This is exciting because, you will explore this concept, so no one comes around in some purely mathematical statement, but we come directly to the most obvious intersection of mathematics, computer science and philosophy. The is the logic.
What is logic?
Mr. Spock would ensure the most appropriate answer, for he knew at least on television forever, which is logical and what is not logical. We have Nichtvulkanier it not so easy. It is said that the logic is the science of rational reasoning. "Gently" is but a first a rather spongy term. What one finds is reasonable or unreasonable for the other. This is due mostly to the content of the messages and the people who taught together this statement. For example, if the environment ministers from the fact that in April this year in Germany, the driest for over a hundred years ago, concludes that the climate is changing in Germany, then a benighted man will hold the (emotionally) to be true. A meteorologist or a climatologist with sufficient experience will dismiss such a conclusion but as unreasonable.
We see people and content make things rather complicated. The logic therefore is interested in neither the one nor the other, but only
for
argument is with the closure by one thing to another. It is not about whether any statements are reasonable, but whether the argument is reasonable. The ancient Greek philosopher Aristotle (384-322 BC). Is widely regarded as the founder of logic, because he was the first who provided comments in writing. From it come many terms that are essential in mathematics. Such as definition
and
evidence. Other philosophers with the wonderful name Diodorus Cronus ,
Philo and Chrysippus of Soli worked out the foundations for a two-valued logic. Other milestones laid George Boole (1847) and Gottlob Frege (1879). We call a logic to classical logic
if it meets the following conditions:
It is bivalent. Divalent means that each statement exactly one truth value, namely either
- "true" or assigned
- false . It is compositional. Compositional means that he d truth value of each compound statement is uniquely determined by the truth values of its sub-statements.
- Now you can compose statements of a compound statement by being linked together by so-called
connectives. Connectives are used much as the negation, implication, equivalence, conjunction and disjunction. To this end, consider for example
atom. So there is in the classical propositional logic indecomposable basic statements that are either true or false and can be linked by connectives together into more complex statements, which are also either true or false. If we want to know whether a statement is true, then we can do, where we divide them into sub-statements, that we know whether they are true or false and then determine the corresponding provision of the connectives between the statements of the truth value of the entire statement.
We move into here in a very interesting area. For it is also clear that mathematics has a lot to do with language
.
0 comments:
Post a Comment