Talk:Proof sketch for Gödel's first incompleteness theorem

message: Good summary and a very clear explanation for the most part, but I got lost here: 
One may now define a set of numbers, which we will denote P, which consists of all numbers in AX (representing axioms) and all numbers which can be derived from them by applying the functions F1, F2,F3... a finite number of times. The set P represents the set of provable statements. That is, the members of P are the Gödel numbers of the provable statements. (Emphasis added)
What inputs are applied to the functions?  The leading comments suggest the numbers in AX. —The preceding unsigned comment was added by 207.171.180.101 (talk) 21:06, 4 April 2007 (UTC)
replies: text: As it stands, this article needs a lot of work, so you may want to consult a print reference for details. The idea is that the functions stand for inference rules, and P is the smallest set of Godel numbers containing the Godel numbers of the axioms and closed under the inference rules. CMummert · talk 21:47, 4 April 2007 (UTC)
thread_title:  Unclear on one part 

message: The list above was all formatting and style issues.  Here are some content issues. They aren't really errors so much as terminology issues. 
#Para. 1: "First, every mathematical statement is written in a fully formal manner," This isn't quite right; arithmetic has a particular formal language and the proof is about particular well formed formulas of that language.
#Paras. 5 and 6.  Although it is possible to replace the deduction rules with partial functions, this just adds complexity to the situation.  The deduction rules are more naturally viewed as relations of some sort than functions.
#Para 7. This terminology is nonstandard.
I will be glad to make these changes myself, once the current discussion at WT:WPM finishes.  The more I read this sketch the more I like it, modulo the terminology. CMummert · talk 20:12, 6 April 2007 (UTC)
thread_title:  Content issues 

message: I offered to provide the coding used by Douglas Hofstadter in his monumental work "Godel, Escher, Bach". Here it is.
:666 0 zero
:123 S successor function
:111 = equality relation
:112 + addition operator:
:236 . multiplication operator
:362 (
:323 )
:212 <
:213 >
:312 [
:313 ]
:262 a variable name
:163 ' prime (used to make more variables)
:161 ^ logical and
:616 v logical or
:633   contains
:223 ~ logical not
:333   there exists
:626   for all
:636 : punctuation
:611 . punctuation
Please correct the above if there are mistakes or omissions. Geometry guy 18:03, 9 April 2007 (UTC)
thread_title:  GEB coding for reference 

message: I started working on the article this afternoon. Terms like "statement" and "statement form" ought to be replaced with the standard terminology - "sentences" or "well formed formulas".  I tried to integrate the coding by Hofstadter from higher on this page, although I'm not sure about the variable x.  I will keep working on other sections.  My goal is to smooth out some of the rough edges and make the article more WP-like.  CMummert · talk 19:39, 18 April 2007 (UTC)
thread_title:   2007-4-18 

message: It seems to me that a minimal suitable formal arithmetic should include additional symbols, such as either a "member of" and "s" for a set, or alternatively a "F" for a formula. Otherwise, it's hard to see how for example the fact that a formula is well-formed can be expressed within the arithmetics, or how the Goedel number of a number can be constructed. For example, it seems like for these tasks, the operation 10^n for every n must be defined (if the Goedel numbers are concatenated in their decimal representation, as in the article), and I can't see how this can be done without additional symbols.
If additional symbols are indeed necessary, this should be noted somewhere. If not, can someone please explain how to construct the above examples without aditional symbols? (with the "member of" and "s" for a set it is clear to me how this can be done, assuming that the deduction rules and some deduction symbols are also incorporated).
Dan Gluck (talk) 21:37, 2 February 2010 (UTC)
thread_title:  Missing symbols?