Features of David Hilbert's Program

Introduction

One of the great Mathematician, whose name was David Hilbert introduced a program in 1890.David Hilbert was a German Scientist and was a good mathematician of his era. It was a great invention. He was famous for his skills of solving difficult statements. According to him, one algorithm can solve all the theorem in the world but he was considered as wrong in the beginning of 20th century by a great Mathematicians, Logician Sir Alan Turing.

David Hilbert Program

One of the main problem on the early 20th century was Paradoxes (It was a severe problem at that time). At that time people didn’t know the term Mathematicians. At that time there was very difficult to solve huge figure of problems and it was the main reason that’s why people thought that all the statements given by the mathematicians were not true.

After that Hilbert was forced to take the statements which were axiomatic. We can say that Axiomatic statements are those statements which we can say that they are true. So, with the use of axiomatic statements he said that his statements are consistence (Consistent statements are those statements which are free of contradiction). He took large number of axioms statements and he told that these statements are consistent which meaning that they are true. By using axiomatic statements he gave his program to solve the problems.

After some time, the theorem which was introduced by David Hilbert was considered as wrong because after some years a scientist whose name was Godel said that this theorem is impossible. Then at that time lots of contradiction were took their place and then Alan Turing and Godel proved that this theorem is impossible.

What was the decision procedure?

I can define word decision procedure that, It is a procedure in which there is two possibilities. By using an algorithm, it can be true or false at a time but not both. Thin procedure tells about that if we have junk of inputs and we have to solve them then there will be two possibilities. The answer of that input by using the algorithm will be in two words. They will be “Yes or No” (True or False, 0 or 1).

The decision by using this procedure is called decision procedure. We can easily understand the term decision procedure by given example.This procedure is very helpful for solving problems. By using this decision procedure we can also solve the problem of any values. This procedure is help to save our time and tells about that the algorithm of this problem can be possible or not.

Let’s take an example in high level language. Suppose that we have lot of inputs and we have to perform some mathematical operation in our program with these inputs. To solve this type of problem we have to make an algorithm which will decide that we can solve these inputs or not and at the end we will get two answers, the answer will be Yes or No. The lines of procedure which define the problem are called decision procedure. Also, there are two other possibilities, one is decidable (if the answer is possible) and other is undecidable (if answer is not possible).

We can say that era of mathematics was going at the peak in 1930 because at that time Alan Turing and Kurt Godel proved David Hilbert wrong who was the great mathematician of his era. This show that this was the starting of modern world, mean to say new emerging field mathematics. The effect of Godel and Turing's leaps forward from the Hilbert program in the 1930s is best comprehended against the foundation of the numerical desire communicated by David Hilbert in the 1920s (however foreshadowed in a popular location that he gave in 1900).

Hilbert, the originator of the "formalist" in Philosophy of Mathematics made in 1921 that specialist's essential target ought to be to build up arithmetic on a predictable establishment of sayings, from which, on a fundamental level, every numerical truth could be derived (by the standard strategies for first request or "predicate" rationale). At that point in 1928 he formulized his Entscheidungs problem or "choice issue" could a viable strategy be contrived which would show in a limited time.

Godel Theorem

In a famous paper published in 1931, Gödel proved that in any true (and consistent) axiomatic theory almost rich to enable the expression and proof of basic arithmetic propositions, it will be possible to construct an arithmetical proposition like G such that neither G, nor its negation, is provable from the given axioms. Hence the system must be incomplete. This is known as Gödel's First Incompleteness Theorem. Moreover it follows from Gödel's reasoning that G must, in fact, be a true statement of arithmetic. He said that arithmetic is incomplete. Godel, after some time proposed 2nd incompleteness Theorem.

Turing's Theorem

Godel incompleteness theorems left the Entscheidungsproblem (Decision Problem). However he showed that any consistent axiomatic system of arithmetic will be leaving a few arithmetical truths unproved, this did not in itself flushed out the existence of some "effectively computable" decision procedure which would in a limited time, reveal may or may not be that any given proposition P was, or was not, provable.

Turing's landmark contribution, in his paper of 1936 "On Computable Numbers, with an Application to the Entscheidungsproblem", was to devise a careful notion of effective computability based on the "Turing Machine". He then thought and proved that there exist problems notably the famous "Halting Problem" for Turing Machines that cannot be effectively computed by this means. That is, he showed that it is impossible to devise a Turing Machine program that can determine easily and completely (and within a finite time) whether or not a given.

This is how Turing proved that Hilbert's Entscheidungsproblem was unsolvable and incorrect. Hilbert's dreams were destroyed, any consistent axiomatic theory, rich to enable the expression and proof of basic arithmetic propositions could be neither complete (as Gödel had shown) nor effectively decided.