Hilbert's second problem

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In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that arithmetic is consistent – free of any internal contradictions.

In the 1930s, Kurt Gödel and Gerhard Gentzen proved results that cast new light on the problem. Some feel that these results resolved the problem, while others feel that the problem is still open.

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Hilbert's problem and its interpretation

In one English translation, Hilbert asks:

"When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. ... But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results. In geometry, the proof of the compatibility of the axioms can be effected by constructing a suitable field of numbers, such that analogous relations between the numbers of this field correspond to the geometrical axioms. … On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms."[1]

It is now common to interpret Hilbert's second question as asking for a proof that Peano arithmetic is consistent (Franzen 2005:p. 39).

There are many known proofs that Peano arithmetic is consistent that can be carried out in strong systems such as Zermelo–Fraenkel set theory. These do not provide a resolution to Hilbert's second question, however, because someone who doubts the consistency of Peano arithmetic is unlikely to accept the axioms of set theory (which is much stronger) to prove its consistency. Thus a satisfactory answer to Hilbert's problem must be carried out using principles that would be acceptable to someone who does not already believe PA is consistent. Such principles are often called finitistic because they are completely constructive and do not presuppose a completed infinity of natural numbers. Gödel's incompleteness theorem places a severe limit on how weak a finitistic system can be while still proving the consistency of Peano arithmetic.

Gödel's incompleteness theorem

Gödel's second incompleteness theorem shows that it is not possible for any proof that Peano Arithmetic is consistent to be carried out within Peano arithmetic itself. This theorem shows that if the only acceptable proof procedures are those that can be formalized within arithmetic then Hilbert's call for a consistency proof cannot be answered. However, as Nagel and Newman (1958:96–99) explain, there is still room for a proof that cannot be formalized in arithmetic:

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