Download Incompleteness and Computability By Richard Zach Pdf book free online – from Incompleteness and Computability By Richard Zach Pdf book; The second aim, that the axiom systems developed would settle every mathematical question, can be made precise in two ways.
In one way, we can formulate it as follows: For any sentence A in the language of an axiom system for mathematics, either A or ¬A is provable from the axioms. If this were true, then there would be no sentences which can neither be proved nor refuted on the basis of the axioms, no questions which the axioms do not settle. An axiom system with this property is called complete. Of course, for any given sentence it might still be a difficult task to determine which of the two alternatives holds. But in principle there should be a method to do so. In fact, for the axiom and
derivation systems considered by Hilbert, completeness would imply that such a method exists—although Hilbert did not realize this. The second way to interpret the question would be this stronger requirement: that there be a mechanical, computational method which would determine, for a given sentence A, whether it is derivable from the axioms or not.
In 1931, Gödel proved the two “incompleteness theorems,” which showed that this program could not succeed. There is
no axiom system for mathematics which is complete, specifically, the sentence that expresses the consistency of the axioms is a sentence which can neither be proved nor refuted.