So instead, I will rephrase and simplify it in the language of computers. Imagine that we have access to a very powerful computer called Oracle. As do the computers with which we are familiar, Oracle asks that the user "inputs" instructions that follow precise rules and it supplies the "output" or answer in a way that also follows these rules. The same input will always produce the same output.
|Published (Last):||1 October 2009|
|PDF File Size:||17.74 Mb|
|ePub File Size:||11.22 Mb|
|Price:||Free* [*Free Regsitration Required]|
So instead, I will rephrase and simplify it in the language of computers. Imagine that we have access to a very powerful computer called Oracle. As do the computers with which we are familiar, Oracle asks that the user "inputs" instructions that follow precise rules and it supplies the "output" or answer in a way that also follows these rules.
The same input will always produce the same output. The input and output are written as integers or whole numbers and Oracle performs only the usual operations of addition, subtraction, multiplication and division when possible. Unlike ordinary computers, there are no concerns regarding efficiency or time. Oracle will carry out properly given instructions no matter how long it takes and it will stop only when they are executed--even if it takes more than a million years.
How would you ask Oracle to decide if N is prime? Tell it to divide N by every integer between 1 and N-1 and to stop when the division comes out evenly or it reaches N Actually, you can stop if it reaches the square root of N. If there have been no even divisions of N at that point, then N is prime. In other words, there are statements that--although inputted properly--Oracle cannot evaluate to decide if they are true or false.
Such assertions are called undecidable, and are very complicated. And if you were to bring one to Dr. Godel, he would explain to you that such assertions will always exist. Even if you were given an "improved" model of Oracle, call it OracleT, in which a particular undecidable statement, UD, is decreed true, another undecidable statement would be generated to take its place.
More puzzling yet, you might also be given another "improved" model of Oracle, call it OracleF, in which UD would be decreed false.
Do you find this shocking and close to paradoxical? It was even more shocking to the mathematical world in , when Godel unveiled his incompleteness theorem. Godel did not phrase his result in the language of computers. He worked in a definite logical system and mathematicians hoped that his result depended on the peculiarities of that system.
But in the next decade or so, a number of mathematicians--including Stephen C. Kleene, Emil Post, J. Rosser and Alan Turing--showed that it did not. Research on the consequences of this great theorem continues to this day. Newman, published in and released in paperback by New York University Press in Read This Next.
GODEL PROOF NAGEL NEWMAN PDF
No member of K is contained in more than two members of L. The analysis consists in noting the various types of signs that occur in a calcu- lus, indicating how to combine them into formulas, prescribing how formulas can be obtained from other formulas, and determining whether formulas of a given kind are derivable from others through explic- itly stated rules gorel operation. To achieve such an understanding, the reader may find useful a brief ac- count of certain relevant developments in the history of mathematics and of modern formal logic. The members of K are not all contained in a single member of L. We repeat that the sole question confronting the pure mathematician as distinct from the scientist who employs mathe- matics in investigating a special subject matter is not whether the postulates he assumes or the conclusions he deduces from them are true, but whether the alleged conclusions are in fact the necessary logical consequences of the initial assumptions. In the second place, the resolution of the parallel axiom question forced the realization that Euclid is not the last word on the subject of geometry, since new systems of geometry can be constructed by using a number of axioms different from, and incom- patible with, those adopted by Euclid. Readers with broader interests, who would like to explore the larger implications of the proof for science or philosophy, may be disappointed that the book ends where it does.
GODEL PROOF NAGEL PDF
In he graduated from the City College of New York, where he had studied under Morris Cohen, with whom he later collaborated to coauthor the highly successful textbook, An Introduction to Logic and Scientific Method Pursuing graduate studies at Columbia University, he received his Ph. In he joined the faculty of Rockefeller University. Nagel was one of the leaders in the movement of logical empiricism, conjoining Viennese positivism with indigenous American naturalism and pragmatism.
ISBN 13: 9780814758373
Being relatively short, this book does not expand on the important correspondences and similarities with the concepts of computability originally introduced by Turing in theory of computability, particularly in the theory of recursive functions, there is a fundamental theorem stating that there are semi-decidable sets sets which can be effectively generated , that are not fully decidable. As expressed beautifully by Chaitin, uncomputability is the deeper reason for incompleteness. And it is precisely by using this fundamental result that Godel could demonstrate his celebrated theorems. Given a formal system such as PA or ZFC, the relationship between the axioms and the theorems of the theory is perfectly mechanical and deterministic, and in theory recursively enumerable by a computer program. Metamathematical arguments establishing the consistency of formal systems such as ZFC have been devised not just by Gentzen, but also by other researchers. For example, we can prove the consistency of ZFC by assuming that there is an inaccessible cardinal.