Everyone says that something is proven, some more ignorant say that something is "scientifically proven." But in reality no notion of what they want to express, sometimes I think they mean, "I say is true and I give my authority for truth value."
But in mathematics the idea of \u200b\u200bthe show is something that is in the very beginning of it. How can we show that something is false or true without any doubts? That problem is what the Greeks began raised. And for that first had to know how we reason. They did not know it, but realized that the argument had certain general laws and tried to machine them.
encode syllogisms Aristotle and Euclid encode geometry. And that's where the matter would be many centuries before someone made a breakthrough in the study of axiomatic reasoning.
(After the Greek humanity was apparently the victim of a virus that affected his brain and went crazy with religion, but of which we are not yet completely free, but since the fifteenth century a larger number people have been saved from infection.)
But the real revolution in this field began in the nineteenth century when other classes were discovered that were not Euclidean geometry.
(To understand your confusion, you must be at that time and wonder how there may be other kinds of points and lines in a unique reality?)
Logic For it was a heavy blow, since she does not reassemble itself from the many paradoxes of the concept of limit in calculus, they left several geometries with a body full axiomatic.
But what really blew it all were the work of Cantor about infinity and numbers after infinite, which led him to set theory based on logic (or rather the encoding of Boole and De Morgan ) other mathematicians continued to make progress together mixing theory with formal reasoning and numbers (such as Peano ) to know it's a show and what are the limits of the demonstrations.
But the whole theory was bringing its own share of paradoxes. The most famous is that of Russell , in which the sets. In which the non-joint (non-members) to itself.
It's like the set of all men is not a man. Luis whole, not me. (People are not sets)
But there are joint devour itself. As the set of all sets, which include yourself to end up creating paradoxes.
Then there are two kinds of sets, the currents that are not included. And they eat themselves, which are included in the set, the logical step was to unite these two types of sets. For example Q, is the set of all ordinary sets.
It is clear that Q is a paradox, and we ask the question "What kind of set is Q, the ordinary or that autodevoran?" We realize that it is neither (to make the test and will see it)
That led to many logicians and mathematicians have tried to make a mathematical free of paradoxes.
The greatest effort is made Russell and whitehead in his masterpiece "Principia mathematics " The idea was to make a style epimenes mathematics without paradoxes, which is something like "this sentence is false." And also the variants that have no apparent paradoxes. As this
A.-The statement that follows is false
b. "The above statement is true.
If we take each one of them isolated, there is no problem in fact commonly used until the problem begins with each one of them says to another, and that's the problem of self-reference.
Russell saw that it was the cause of all evil and so his attempt to make a free math them
The way to avoid formation of paradoxes in set theory and that of the numbers was quite elegant, but too artificial. It was the theory of types.
This is the simplest set consisted only of specific things that could not be subdivided into sets, the following could only be formed by simple sets, and continues that way. Clearly
seen thus avoiding sets as Q, since none can belong to himself and that is prohibited.
But Epimenides paradox style continued as
And when he began publishing his work a lanky young mathematician realized that there was something terribly wrong with all this and set out to prove it was Kurt Godel, as I have seen in several places referred to his discovery, but often not well explained. Or make it very lightly, so that only ends in a cartoon.
The incompleteness theorem is known by few people, despite their importance in almost all areas, it is extrapolated that shamelessly, I do it myself. But I will try to formally explain what are the limits you set.
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