Re: Godel's comments about the "true reason" for incompleteness

djrt20@xxxxxxxxxx writes:

"The true source of the incompleteness attaching to all formal systems
of mathematics, is to be found---as will be shown in Part II of this
essay---in the fact that the formation of ever higher types can be
continued into the transfinite (c.f., D. Hilbert, 'Über das
Unendliche', Math. Ann. 95, p. 184), whereas in every formal system at
most denumerably many types occur. It can be shown, that is, that the
undecidable propositions here presented become always become decidable
by the adjunction of suitable higher types. A similar result also
holds for the axiom system of set theory."

This comment by Godel has me confused, first of all by what he means
by "true source". Isn't his proof and later refinements/
generalisations of it a "true source" for incompleteness?

They show that the phenomenon is there, but if we are looking for
an explanation of why the phenomenon occurs, we can look elsewhere
also.

Also, I was
under the impression that the whole point of Godel's theorem is that
any kind of proof procedure or list of proof procedures that you can
*even indicate* will not be able to decide all mathematical
propositions. It sounds as if he is saying "ah, we can just continue
adjoining higher types in such and such a manner, and eventually
arbitrary statements become decidable (i.e. for any statement, it
eventually becomes decidable)". I thought the whole point of Godel was
that even if you spent a billion years outlining precisely a method of
coming up with formal systems, there would still be propositions that
could not be resolved by any of those formal systems.

well yes, but what if there is a possible transfinite way of adding in
new properties, which does not correspond to a formal system? I take
it that this is what he is referring to -- others may know the context
better. The "undecidable properties here presented" are formalised
consistency statements, which Gödel argues are in fact true, so by
using a reflection principle we can strengthen the given theory (and
iterate this transfinitely, but not effectively, to get a complete,
therefore decidable theory).

Feferman's "Turing in the land of O(z)" is a good exposition.


--
Alan Smaill
.

Relevant Pages

  • Re: Godels comments about the "true reason" for incompleteness
    ... of mathematics, is to be found---as will be shown in Part II of this ... by the adjunction of suitable higher types. ... any kind of proof procedure or list of proof procedures that you can ... formal system whereas Pis not enumerable. ...
    (sci.logic)
  • Re: Godels comments about the "true reason" for incompleteness
    ... of mathematics, is to be found---as will be shown in Part II of this ... by the adjunction of suitable higher types. ... any kind of proof procedure or list of proof procedures that you can ... formal system whereas Pis not enumerable. ...
    (sci.logic)
  • Re: Godels comments about the "true reason" for incompleteness
    ... of mathematics, is to be found---as will be shown in Part II of this ... by the adjunction of suitable higher types. ... any kind of proof procedure or list of proof procedures that you can ... formal system whereas Pis not enumerable. ...
    (sci.logic)
  • Re: Godels comments about the "true reason" for incompleteness
    ... of mathematics, is to be found---as will be shown in Part II of this ... by the adjunction of suitable higher types. ... he is saying that a system or proposition is manifested by a higher system that manifests it. ... As any member of all systems must have a higher system if that member is to be complete, it follows that the totality is incomplete. ...
    (sci.logic)
  • Re: Godel and Kant, and incompleteness
    ... formal systems of mathematics is that the formation of ever ... higher types can be continued into the transfinite." ... 'Transfinite' need not mean 'ad infinitum'. ...
    (sci.logic)