On Wed, Jul 25, 2012  Stephen P. King <stephe...@charter.net> wrote:

  > 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 catastrophic.

  John K Clark

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