( A Part 1 - Part 3 A )
Dialectics Syntax / Semantics
Gödel's theorems, published in 1931, are part of a long debate on the fundamentals of mathematics that began in 1872 with the discovery, by Cantor, the transfinite, which had been boosted since 1902 with the discovery of Russell's Paradox.
The battlefield of the controversy was nothing less than infinity. Constructivist school, led by LEJ Brouwer, argued that the introduction of the actual infinite in mathematics was absurd and unjustified and that the theory of transfinite Cantor was just a meaningless word game. The only valid mathematical objects, held this school are those that can be constructed mechanically in a finite number of steps. For example, they could not speak of all the natural numbers, but finite and an ever increasing numbers are calculated individually. The statements speak of infinite totalities, for the constructivists were meaningless.
By 1920 the controversy involved in David Hilbert, who in a series of papers published over ten years, he proposed what is today known as the Hilbert Program and essentially , had the requirement finitude and constructive objects mathematical to mathematical reasoning.
More precisely, Hilbert proposed the creation of a new science that he called meta- . This science would aim to verify the validity of mathematical reasoning. To avoid controversy (and to ensure that no paradoxes arise) this science would be purely finitary: the meta-treat statements and mathematical reasoning as if they were just meaningless strings of symbols to which mechanically manipulate a finite amount of steps.
has In some texts the disclosure that the Hilbert program sought to reduce mathematics to a game of meaningless symbols, has also said that for Hilbert the concept of "mathematical truth" did not exist. Nothing more false. Hilbert, we understand, was primarily a research mathematician (the best of their time) and it is impossible, unthinkable, I could think so. Hilbert attributed these traits, not mathematics, but metamathematics.
Mathematics works at a semantic level, full of meanings. The mathematician, on the day to day, always working, believe, conjecture, showing and suffering, as if what he had in hand were real objects.
metamathematics based on the idea of \u200b\u200bHilbert, who works at the syntactic level, provides methods to check the arguments, that the mathematician has obtained as a final result of his creative work, are correct. To do this check the reasoning would be loaded on a computer that would verify whether the reasoning is valid or not. Hilbert, of course, not talking about computers, but what I said in the previous sentence reflects the gist of Hilbert: the validity of the argument is checked by mechanical manipulation of symbols made in a finite amount of steps (the verification of reasoning, it was obtained).
Specifically, the proposed Hilbert program to a set of axioms for arithmetic (1) that met four conditions:
1. The system should be consistent (ie, there should not be a statement P such that P and its negation be simultaneously shown).
2. The validity of any demonstration must be verifiable by mechanical manipulation (syntax) in a finite number of steps.
3. Given any statement P, either he or his denial should be demonstrable.
4. The consistency of the axioms must be verifiable fnita mechanically in an amount of steps.
Note that these conditions are not mathematics but metamathematics. If these four conditions could be met then the notion of "provability" (syntax) and "truth" (semantics) could be considered equivalent. But Gödel's theorem showed precisely that these four conditions can not be fulfilled at once. Are met and 1 2 3 then it is false and 4 , if the system is reasonably powerful, it is impossible.
In the next chapter that they wish to say "reasonably strong" and begin to analyze the proof of Gödel.
continued ...
(1) Note: Arithmetic is the theory that talks about the addition and multiplication of natural numbers. Hilbert believed that this was the fundamental theory of mathematics (and not set theory).
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