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

djrt20@xxxxxxxxxx wrote:
"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? 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.

The 'true source' he refers to tackles the classical problem of totalising an infinite series. His 'proof' is the mathematical/logical expression, clarification or presentation of that problem, and not so much a proof of it.

Basically, he is saying that a system or proposition is manifested by a higher system that manifests it. However, this system itself must be incomplete except when it is manifested by a yet higher system.
As any member of all systems must have a higher system if that member is to be complete, it follows that the (transfinite) totality is incomplete.

Here, Godel assumes that a manifesting system can, through a self-identification, act as an object for a yet higher system, and on into the transfinite. But it is arguable whether a system that manifests objects can also act as an object in a higher system.

Godel's description of the problem is predicated on the equivalence of objects with their manifesting conditions or systems. This leads to a problematic notion of totality. His proof plays out a circularity in that by adopting a mathematical proof he could reach no other end than the one he describes, but the picture he proposes is not ontologically accurate even if the limited notion of totality that he presents drives or underpins his proofs.
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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 ... which does not correspond to a formal system? ...
    (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: 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)
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