Re: Godel Contradiction


IF T* doesn't include G(T) as an axiom then T* would have to include
other axioms that would allow the derivation, in T*, of E [ G(T) ],.
And so the proof G(T) would rest on the acceptance of these other
axioms instead.

Correct.

Now, here's the juicy part: Our T* can be proven in a theory as
limited as primitive recursive arithmetic (PRA) and using only
intutionistic logic. (I don't know the details of proving that - it
involves some fairly sophisticated "coding" - but you can investigate
that matter in literature.)

if you don't know the details of proving that then on what basis do
you assert it as true.?


In other words, we can prove the
incompleteness theorem in a theory T* that pretty much assumes nothing
more than algorithmic arithmetic done constructively.

now HERE'S where one of us is getting confused:
BECAUSE...
IF T* is PRA plus a few other axioms, then by your logic, T* can prove
E [ G(T*) ] since a subset (PRA) of T* can.

and that just sounds like self-contradictory gibberish to me.

i.e what if PRA is a subset of T* ?




Moreover, we may ask you, "From what axioms do you ordinarily accept
as proving any of various mathematical theorems, ranging from "1+1=1"

interesting new theorem there MoeBlee !

So why in the world would you accept a
whole bunch of mathematical theorems as proven but not the
incompleteness theorem that can be proven from an even WEAKER set of
assumptions than those used to prove all the mathematical theorems you
accept?

That the incompleteness theorem CAN "be proven from an even WEAKER set
of
assumptions than those used to prove all the mathematical theorems I
accept" I only have so far by your ipse dixit and the reason I don;t
yet accept it is because of a seeming contradiction in T* proving the
incompleteness theorem for itself using that subset of itself called
PRA.

.

Relevant Pages