According to book Georg Cantor: His Mathematics and Philosophy of the Infinite - Joseph Warren Dauben , David Hilbert claimed that the axiomatic method to real number system is more secure than the constructive method(genetic method) .
The axiomatic method described as
These axioms included those for the familiar operations of arithmetic, commutative and associative laws, order relations, an axiom of continuity (the Archimedean Axiom), and a completeness axiom, which stated that from the axioms listed no other numbers could be generated in addition to the primitive elements, or numbers, assumed at the outset. From these it was then necessary to establish that the system of primitive concepts and axioms were together consistent and complete, that no contradictions could result from any combination of axioms, and that the system of axioms was sufficient to prove all theorems possible concerning the real numbers.
The constructive method(genetic method)
Beginning from the sequence of natural numbers generated successively from the unit, the integers 1, 2, 3, .. . , eventually led to negative numbers, fractions, and ultimately to real numbers by successively extending the simple concept starting with the natural numbers. The logical base of this approach is some series of assertions concerning the natural numbers only, for example, Peano's axioms. All the other numbers are constructed. Hilbert called this approach the genetic method (he may not have known Peano's axioms at this time but he knew other approaches to the natural numbers).
Hilbert grants that the genetic method may have pedagogical or heuristic value, but believed that it could never ensure complete logical certainty. To satisfy the demands of absolutely certain knowledge, only an axiomatic method would suffice.
- Why the constructive method(genetic method) could never ensure complete logical certainty? What flaws inherently within this method?
- Why the axiomatic method could satisfy the demands of absolutely certain knowledge?
I want more explanation, thanks !