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 -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
