I'll talk about an interesting theory today.
Deficiency theory;
Let's take a closer look at this theorem of Austrian mathematician Kurt Gödel, which he unveiled in 1931.
In mathematics, the anxiety of having the danger of being able to math the mathematics of self-reference paradoxes forced bertrand russell and alfred north whitehead to construct a mathematics that would not create a cyclicality.
especially in the theory of the clusters, the 'self-sucking cluster', 'the cluster of all the clusters', leading to paradoxes.
The result of this heavy work was born principia mathematica in 1910 - 1913.
mathematics was freed from its references, liberated. happy ending…
// how beautiful life does not end with happy ends like freaky episodes, let's break up now, the show is over.
About twenty years after the birth of the machine principia mathematica, a young mathematician named kurt gödel discovered the technique of 'gödel'.
The Höfstadter describes gödel quantification as "a mapping in which the long linear arrangements of symbol sequences on any formal string are fully reflected by mathematical relations between certain integers".
The biggest impact of the Gödel quantification was that "your mathematics can talk about itself".
any statement about a mathematical sequence can be examined in number theory;
that the expressions can be transformed into relations of numbers and numbers, allowing the expressions themselves to enter the mathematical world.
such as 'I can be proved in I principia mathematica' or 'I can not be proved in principia mathematica', proved to be provable in number theory - that this mathematics can speak about itself in its own language.
mathematics is self-perception and self-awareness.
but the use of this aesthetic and powerful transformation on mathematics, especially in the second form, suggesting its unprovenability, led to surprising results.
the entrenched paradox was in the middle of his mathematics;
(The epimenides paradox) that "all the entrances are liars."
principia mathematica will use typographical number theory (tnt), which is more general, rather than special case;
The proof that the paradoxical form of 'I can not be proved in TNT' is one of the theorems of principia mathematica is a result that can not be readily absorbed by classical reasoning.
Gödel's second theorem says that the absence of this form (~ g say) creates the same contradiction.
In this case, it can not be decided whether or not g is itself or not.
such that coherent sequences such as TNT have theorems that can not be decided on their correctness.
Since there is no method of catching the theorems that can not be decidable, the number of the theorems that can not be decided on a consistent string is unknown.
(which evokes the astonishment of realizing that real numbers reside in a space of incredible size, assuming that the spaces between natural numbers on the number line are filled with rational numbers).
even according to GÖDEL, this deficiency has to be found in all coherent sequences.
The Höfstadter has very well exemplified this state of necessity in the dialogues of the tosbağa and the akhilleus in his book:
"For every recorder he has a record he can not play." Tosbag, Mr. Crab gives a name label "I can not play in these recordings".
This record, arranged in such a way as to resonate and dissipate the mechanism when the recorder starts to play, is the hall of godel itself. If the player tries to play the information coded on the plate as 'consistent', he prepares his own end.
as one of the ways of getting rid of it, it seems to make the mechanism inconsistent.
but in this case the non-g type 'innocent' plaques can not be played correctly.
this is much worse than in the first case, pink floyd is another sound from the grass. It is vulgar.
a consistent system, principia mathematica, has taken its course from this necessity, and it has been understood that there are inevitable shortcomings in the Gödel '
there is no need to worry, such crises are an opportunity for the opening of new gates in the world of science, the expansion of science.
the hofstadter is a number one that is to be used after classical logic is first falsified, as if people understood that 2 could not be expressed as the ratio of the two integers of the square root, or that they had to accept the usefulness and consistency of the complexity when it was thought that ' 'supernatural numbers' can be used to express these paradoxical situations.
but these numbers are infinitely large integers used in inferences to be made on undecided theorems such as g and ~ g.
the theorem is important, if we are strong the same enemies like roger penrose and strong ai fans like douglas hofstadter, it transforms us into an insignificant one. so powerful, so alone ...
Thank you for reading%)>
Nice post