Re: 3rd paradox in Godels incompleteness theorem that makes it invalid

On Oct 19, 5:36 am, "elsiemelsi" <cyprin...@xxxxxxxxxxxxxxx> wrote:
In other words PM is included in those systems which have undecidablity

Correct.

Thus we have the paradox

You WOULD NOT KNOW what a PARADOX
was if one walked up to you and smacked you upside the head.

that while PM is used to find if a formal system
is undecidable

PM by itself IS NOT used for that in this proof.
It could've been, but that is more a consequence of
the discovery of the proof. SOME OTHER THINGS ABOVE AND
BEYOND PM are used to prove a result ABOUT PM (and other
systems with some features IN COMMON *with* PM) in this proof.

it is undecidable itself

This is entirely true, but THERE IS NOTHING PARADOXICAL
about it. One doesn't go dismissing WHOLE SYSTEMS as
"undecidable": LOOK AT THE TITLE of the work:
"On Formally Undecidable PROPOSITIONS IN PM etc.."
The work ADVERTISES IN ITS TITLE that PM contains undecidable
propositions! But containing SOME undecidable propositions has
NO effect whatsoever on the OTHER propositions that ARE not
only decidABLE, but decidED, and decided positively, i.e., PROVEN!

i.e.
hence in every formal system which satisfies assumptions 1 and 2 [ from P

which uses system PM]

You keep inserting this likle that insertion is going to make it
relevant.
There are PLENTY OF SYSTEMS satisfying assumptions 1 and 2 that
have NOTHING WHATSOEVER TO DO with P or PM.
This theorem APPLIES TO THEM, TOO. All these systems contain
undecidable
propositions and all of them could've been used to prove this theorem.
The theorem that IF one of these systems is consistent, then at least
one
of its sentences -- namely, its own consistency sentence -- is
undecidable,
IS NOT one OF the undecidable sentences these theories. THAT
sentence,
UNLIKE the consistency setnence, IS provable, and true, in all these
theories
and all their models.

and is w - consistent there exist undecidable
propositions

In other words the very system which is used to find undecidability is
included in the set of undecidable systems

Yes, it is, but there is absolutely nothing paradoxical about that,
because throughout each and every one of these systems, the
PROPOSITION that is this theorem IS NOT included among the
undecidable PROPOSITIONS of the theory. The consistency sentence
is included among the undecidable propositions, BUT THAT'S JUST
WHAT THE THEOREM SAYS.

Thus we have the situation overall that clearly Godel is in paradox

Maybe you should learn the definition of the word "paradox"
before you continue.

.