PPT Slide
The Hilbert Problems contd.
Many of the problems have since been solved, and each solution was a noted event.
Hilbert second problem asked whether it can be proved that
that the axioms of arithmetic are consistent (that is, that a finite number of logical steps based on them can never lead to contradictory results). Godel’s solution: you cannot tell, because propositions can be formulated that are undecidable within the axioms of arithmetic!
Example: By definition, x2=x.x, x3=x.x.x, and so on. Now what does x3/5 mean? Can we be certain that the meanings that we have given to fractional and non-rational exponents are always consistent with the natural meaning of positive integral exponents? That is the nature of Hilbert second problem.