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Concept
Hilbert's second problem
Summary
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 the arithmetic is consistent – free of any internal contradictions. Hilbert stated that the axioms he considered for arithmetic were the ones given in , which include a second order completeness axiom.
In the 1930s, Kurt Gödel and Gerhard Gentzen proved results that cast new light on the problem. Some feel that Gödel's theorems give a negative solution to the problem, while others consider Gentzen's proof as a partial positive solution.
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."
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.
Official source
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