Hilbert's program

In mathematics, Hilbert's program was an important idea created by the German mathematician David Hilbert in the early 1920s.

At that time, mathematics had some problems. New ways to explain the basics of math sometimes led to mistakes. Hilbert wanted to fix this. He planned to create a strong set of rules, called axioms. These rules would be the base for all math. He believed these rules would not lead to mistakes. He also thought more complex math, like the study of real numbers, could be proven correct using these rules.

However, in 1931, a mathematician named Kurt Gödel showed that Hilbert's program would not work for important parts of math. Gödel proved that in any system with basic rules that can handle arithmetic, there will always be true statements that cannot be proven with those rules. He also showed that such a system cannot prove that it itself is correct. This was a big discovery. It meant Hilbert's hope of proving all of mathematics was not possible.

Statement of Hilbert's program

The main goal of Hilbert's program was to make strong, safe rules for all of mathematics. This meant writing every math idea in a clear way, using special steps to work with them.

Hilbert wanted proofs to show that these rules were complete, meaning every true math idea could be shown. He also wanted to show they were consistent, meaning no mistakes or opposite answers could happen. He hoped to find a way to decide if any math statement was true or false using a set of steps.

Gödel's incompleteness theorems

Main article: Gödel's incompleteness theorems

Kurt Gödel showed that many ideas from Hilbert's plan for mathematics would not work as Hilbert hoped. Gödel's work proved that a mathematical system strong enough to handle basic number operations cannot prove that it itself is correct. This means it is not possible to capture all true math facts in one system, because some true facts will always be left out. Also, a system like Peano arithmetic cannot prove its own correctness.

Hilbert's program after Gödel

Many areas of mathematical logic, like proof theory and reverse mathematics, continue ideas from Hilbert's original plan. Much of Hilbert's plan is still useful, even with some changes.

Although we cannot organize all of mathematics into one system, we can organize nearly all the math that people use. For example, Zermelo–Fraenkel set theory, together with first-order logic, is a good way to structure most of today's mathematics. While we cannot prove completeness for systems that include basic number theory, we can prove completeness for many other important systems. One example is the theory of algebraically closed fields with a specific characteristic.

It is hard to know whether there are simple proofs showing that strong theories are consistent. Most experts think simple mathematics is included in basic number theory, and with this view, it isn't possible to give simple proofs for very strong theories. However, some believe simpler methods can be used. Later, Gentzen provided a consistency proof for basic number theory. The only part of this proof that might not be simple was a type of reasoning called transfinite induction up to the ordinal ε₀. If this reasoning is considered simple, then we can say there is a simple proof showing that basic number theory is consistent. Other mathematicians have also provided consistency proofs for stronger systems.

While there is no universal method to decide the truth of statements in basic number theory, there are many interesting theories for which such methods have been developed. For example, Tarski created a method to decide the truth of any statement in analytic geometry. With the Cantor–Dedekind axiom, this method can also decide the truth of statements in Euclidean geometry.