Editing Hilbert's program (section)

==Hilbert's program after Gödel==
Many current lines of research in [[mathematical logic]], such as [[proof theory]] and [[reverse mathematics]], can be viewed as natural continuations of Hilbert's original program. Much of it can be salvaged by changing its goals slightly (Zach 2005), and with the following modifications some of it was successfully completed:
*Although it is not possible to formalize '''all''' mathematics, it is possible to formalize essentially all the mathematics that anyone uses. In particular [[Zermelo–Fraenkel set theory]], combined with [[first-order logic]], gives a satisfactory and generally accepted formalism for almost all current mathematics.
*Although it is not possible to prove completeness for systems that can express at least the Peano arithmetic (or, more generally, that have a computable set of axioms), it is possible to prove forms of completeness for many other interesting systems. An example of a non-trivial theory for which [[Complete theory|completeness]] has been proved is the theory of [[algebraically closed field]]s of given [[Characteristic (algebra)|characteristic]]. 
*The question of whether there are finitary consistency proofs of strong theories is difficult to answer, mainly because there is no generally accepted definition of a "finitary proof". Most mathematicians in proof theory seem to regard finitary mathematics as being contained in Peano arithmetic, and in this case it is not possible to give finitary proofs of reasonably strong theories. On the other hand, Gödel himself suggested the possibility of giving finitary consistency proofs using finitary methods that cannot be formalized in Peano arithmetic, so he seems to have had a more liberal view of what finitary methods might be allowed. A few years later, [[Gerhard Gentzen|Gentzen]] gave a [[Gentzen's consistency proof|consistency proof]] for Peano arithmetic. The only part of this proof that was not clearly finitary was a certain [[transfinite induction]] up to the [[ordinal number|ordinal]] [[Epsilon number|ε<sub>0</sub>]]. If this transfinite induction is accepted as a finitary method, then one can assert that there is a finitary proof of the consistency of Peano arithmetic. More powerful subsets of [[second-order arithmetic]] have been given consistency proofs by [[Gaisi Takeuti]] and others, and one can again debate about exactly how finitary or constructive these proofs are. (The theories that have been proved consistent by these methods are quite strong, and include most "ordinary" mathematics.)
*Although there is no algorithm for deciding the truth of statements in Peano arithmetic, there are many interesting and non-trivial theories for which such algorithms have been found. For example, [[Alfred Tarski | Tarski]] found an algorithm that can decide the truth of any statement in [[analytic geometry]] (more precisely, he proved that the theory of [[real closed field]]s is decidable). Given the [[Cantor–Dedekind axiom]], this algorithm can be regarded as an algorithm to decide the truth of any statement in [[Euclidean geometry]]. This is substantial as few people would consider Euclidean geometry a trivial theory.