On 7/25/2012 4:18 PM, John Clark wrote:
On Wed, Jul 25, 2012 Stephen P. King <stephe...@charter.net
> This lack of uniqueness [of Godel numbering] is a huge weakness!
A huge weakness in regard to what? It certainly does not weaken
Godel's proof! A Godel number is just a name given to a well formed
symbol or sentence in a formal logical system, any statement in a
logical system has a unique number. It turns out that there are an
infinite number of ways to assign a unique number (name) to every such
symbol or sentence, but Godel only needed one way to prove what he
wanted to prove so he picked a method that he happened to like and was
in his opinion easier to use than most. If you are uncomfortable with
his way for some reason use your own method for assigning numbers to
statements, the results will be the same. If there is more than one
way to prove something it does not make it less certain.
What Godel then did was find a way, in any logical system that is
powerful enough to do arithmetic, to carefully construct a sentence
that says in effect "the formula that has the Godel number G can not
be proven in this system"; and then showed that the Godel number of
that very sentence is G. Incredibly he found a way for a formula to
talk about itself, in this case it is saying "I can not be proven", so
if you prove it then your logical system is inconsistent, you can
prove things that are not true, and if you can't prove it then it's
incomplete. Incomplete is unfortunate but inconsistent would be
John K Clark
It is a weakness of the "1p plural" idea as Bruno defines it.
"Nature, to be commanded, must be obeyed."
~ Francis Bacon
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