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

On Mar 12, 1:37 pm, LauLuna <laureanol...@xxxxxxxx> wrote:
On Mar 12, 4:39 am, Newberry <newberr...@xxxxxxxxx> wrote:





On Mar 11, 4:30 pm, LauLuna <laureanol...@xxxxxxxx> wrote:

On Mar 11, 6:47 pm, djr...@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.

This is an interesting topic. Among other reasons because Gödel never
gave a proof, for all I know, although he mentioned the point later
again.

Gödel stated that higher types with the corresponding comprehension
axioms render the undecidable sentences decidable (though, of course,
there are also undecidable sentences in the resulting systems).

I wonder how exactly this is related to the following:

1. Only an enumerable number of sets of naturals are definable in a
formal system whereas P(N) is not enumerable.
2. Gödel stated elsewhere that the ultimate cause of arithmetic
incompleteness is the fact that arithmetic truth is not arithmetically
definable (Tarski 1933); it's natural to presume he had in mind some
relationship between the two phenomena.

The reason for incompleteness is that bivalence is inadequate.- Hide quoted text -

- Show quoted text -

How so?

For the same reason "this sentence is not tue" has neither of the two
values. Furthermore the so called "vacuously true sentences" are
neither true nor false. Goedel's formula is vacuously true

~(Ex)(Ey)(Pxy & Qy) (1)

Let m be the Goedel number of (1), and Qy is satsfied only by y = m.
If

~(Ex)Pxm

then (1) is vacuously true. If vacuously true is actually not true
(and not false either) then (1) does not have a truth value. This of
course is not the case in classical logic. A non-bivalent logic is
required, and that is basically what Goedel's theorem tells us.

.

Relevant Pages

  • Re: Godels comments about the "true reason" for incompleteness
    ... by the adjunction of suitable higher types. ... there are also undecidable sentences in the resulting systems). ... formal system whereas Pis not enumerable. ... Furthermore the so called "vacuously true sentences" are ...
    (sci.logic)
  • Re: Godels comments about the "true reason" for incompleteness
    ... 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. ... Furthermore the so called "vacuously true sentences" are ...
    (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. ... any kind of proof procedure or list of proof procedures that you can ... formal system whereas Pis not enumerable. ...
    (sci.logic)