Editing Hilbert's second problem (section)

== Hilbert's problem and its interpretation ==

In one English translation, Hilbert asks:
<blockquote>
"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."<ref>{{harvtxt|American Mathematical Society|1902}}, translated by M. Newson. For the original version, see {{harvtxt|Hilbert|1901}}.</ref> </blockquote>

Hilbert's statement is sometimes misunderstood, because by the "arithmetical axioms" he did not mean a system equivalent to Peano arithmetic, but a stronger system with a second-order completeness axiom. The system Hilbert asked for a completeness proof of is more like [[second-order arithmetic]] than first-order Peano arithmetic.

As a nowadays common interpretation, a positive solution to Hilbert's second question would in particular provide 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 are 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 [[finitism|finitistic]] because they are completely constructive and do not presuppose a completed infinity of natural numbers.  Gödel's second incompleteness theorem (see [[Gödel's incompleteness theorems]]) places a severe limit on how weak a finitistic system can be while still proving the consistency of Peano arithmetic.