Hilbert's Second Problem - Hilbert's Problem and Its Interpretation

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."

It is now common to interpret Hilbert's second question as asking in particular for a proof that Peano arithmetic is consistent.

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.