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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.

Questions :

  1. Why the constructive method(genetic method) could never ensure complete logical certainty? What flaws inherently within this method?
  2. Why the axiomatic method could satisfy the demands of absolutely certain knowledge?

I want more explanation, thanks !

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  • $\begingroup$ Don’t know about Hilbert’s justifications, but, roughly speaking, there’s no need to choose the only one of the two. Whether you do or don’t have an axiomatization of some theory (e. g. theory of Dedekind-complete fields), you still have to show the theory has models; if you have defined some object, you still may show it is a model of some theory, and then apply your knowledge of that theory. $\endgroup$ – arseniiv Dec 21 '17 at 18:52
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    $\begingroup$ "the book"? Is this the one with the proofs that Erdős was talking about? $\endgroup$ – Asaf Karagila Dec 22 '17 at 10:33
  • $\begingroup$ @AsafKaragila book books.google.com/… $\endgroup$ – iMath Dec 22 '17 at 11:58
  • $\begingroup$ Yes. My point is that the name of the book should be clear without going off the site. For example, by giving the title of the book, or something "sufficiently recognizable". $\endgroup$ – Asaf Karagila Dec 22 '17 at 11:59
  • $\begingroup$ (Also when citing paragraph, it's a good idea to give as exact and accurate citation, including page numbers and if possible paragraph numbers.) $\endgroup$ – Asaf Karagila Dec 22 '17 at 12:00
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See: David Hilbert, On the foundations of logic and arithmetic (1904).

The issue is with the consistency of arithmetic and analysis:

Having thus established a certain property for the axioms adopted here, we recognize that they never lead to any contradiction at all, and therefore we speak of the thought-objects [Gedankending] defined by means of them [the basic concepts of the theory, used in the axioms], as consistent notions or operations, or as consistently existing.

See also: David Hilbert, Axiomatic Thinking (1918):

Corresponding to this demand in the Grundlagen der Geometry, [Foundations of Geometry] I proved the consistency of the erected axioms, in which I showed that each contradiction in the deduction from the geometric axioms must necessarily be discernible in the arithmetic of the real number system as well.

Also the question of the consistency of the axiomatic system for real numbers is reduced, through the use of set theoretic concepts, to the same question for integers. This is the merit of the theories, by Weierstrass and Dedekind, of irrational numbers. Only in two cases, namely if it is a question of the axioms of integers themselves, and if it is a question of the foundation of set theory, this mode of reduction to another specific field of knowledge is manifestly impracticable, since beyond logic there is no more discipline to which an appeal could be lodged.

And see also the Lecture delivered by Hilbert before the International Congress of Mathematicians at Paris in 1900, with the exposition of the well-known list of problems. Specifically, see:

2. The compatibility of the arithmetical axioms

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  • $\begingroup$ someone said on this particular issue Hilbert was (probably) wrong. Godel and others (and a couple of world wars) have shown that the supposed sure and certain foundations aren't. $\endgroup$ – iMath Apr 5 '18 at 10:31

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