The Turing Machine

Imaginary device which formalized the fundamentosda computer science. Turing showed that computation of the operations of reading, writing and deleting binary symbols could be satisfied by a machine that contained a tape of unlimited length, with length fields defined on it and a device with a finite number of states conducting operations. See more about the next model: aturingmachine.com

Lego model

Regarding the model of the previous example, its implementation may be easier, but will require a lot of work to perform the operations of reading, writing and deleting and perform some simple algorithms such as summation and subtraction. In an address given for details of construction and contact the authors:
legoofdoom.blogspot.com

The machine concept

This is a device imaginary revolutionary created by Turing to 24 years of age. The Turing machine has formed the basic structure to support the modern computer science and computability. His invention was responsible, years later, the recognition of the international scientific community, naming Turing with the symbolic title "father of computing".

Computable Numbers

The theory was published in 1936, in the article "On Computable Numbers, with an Application on the Entscheidungsproblem," in response to treatment of the decision problem, formulated by Hilbert. Despite the Turing machine has not been physically implemented in full by its author, the computational process has been mathematically demonstrated and proven in the article.

The mathematical basis

The Incompleteness Theorems of Gödel, 1931, represent the end of the romantic age of mathematics.
Before Gödel, David Hilbert proposed that every mathematical problem is soluble.
"If there is a problem, find the solution you can find it just thinking, because there ignorabimus in mathematics".

Consistency

An axiomatic theory is said to be consistent when it does not derive a contradiction, ie, are not derived from a proposition and its negation. For example, the theory of algebras Boole can be considered consistent because Own a model finite. When rules used correspond classical logic, if the theory is inconsistent, from a contradiction can be derived any statement, trivializing the system.

Completeness

An axiomatic theory is called complete if for every proposition P theory (formula without free variables), or can be deduced from P and may be deducted or the negation of P. Examples of complete mathematical theories we mention the theory of algebraically closed body feature sets and the theory of Boolean algebras without atoms.

The Hilbert propose

Axiomatize the whole body of mathematical knowledge (including aiming to prove that every mathematical problem was soluble).

Prove, by means strictly finitary, that is consistent axiomatic desired.

Gödel on the Scene

In 1931 Kurt Gödel said shyly: "You can actually display sentences that are true but unprovable in the formal system of mathematics".

O único dos presentes que imediatamente compreendeu a revolução que se iniciava foi John von Neumann.

Mathematicians have just been expelled from paradise ...

Jon Von Neumann

Hungarian mathematician and was a naturalized American. Mathematics major contributions:

• Functional Analysis
• Ergodic Theory
• Statistics
• Number Theory
• Formalization of Computer