Talk:Gödel numbering

message: \forallx, ¬R(v,x) Can also be interpeted to mean "The negation of a proposition of type v can be proved", yielding:
x=(0,1,¬, ask)
R(0,0)=¬0 = ask 1
R(1,1)=(¬0)0 = ask 0
R(1,0)=0
R(0,1)=1
Hackwrench 04:29, 12 November 2005 (UTC)
thread_title: Negation

message: I don't understand the formal account of the proof.  Surely the first statement in fact means simply "v cannot be proved"?  Where does this "type" thing come into it?
Then does not the next statement say "it cannot be proved that v cannot be proved"?  Which doesn't condense into anything, I don't see how the "v" can disappear out of the equation.  It seems to say nothing more than that "v" is undefined, which seems reasonable, since we didn't define it (although just because a statement says something doesn't make it true).
82.41.211.70 22:18, 25 April 2006 (UTC)
thread_title: Informal proof doesn't make sense

message: I'm not convinced that the  notice is helpful, nor that this article belongs in :Category:Wikipedia articles needing clarification which is the result of this tag.
Certainly the subject is a confusing one, particularly to non-specialists. Even many specialists are still trying to figure out exactly what the consequences are, both for Mathematics and for Formal logic. I'm not familiar with the consequences for computing, I even wonder whether that's at least partly an urban myth, based on the (misleading IMO) connections to Alan Turing and Turing machines.  
But if the subject is a newsy and confusing one on which many are already misinformed before they come here, that's not the fault of the article.
What are the specific ways in which the article can be improved? Bear in mind that if it were to make this subject look simple to a non-specialist, that would be very good evidence that the article was then inaccurate. Andrewa 20:33, 24 October 2006 (UTC)
thread_title: Confusing

message: The current definition (without the vandalism) is nonsense.  Does anybody have a definition that satisfies these characteristics?
* Correct
* More general than just formulas of an effective first-order language
* Used in print
I would remove the Definition section and replace it with a discussion of the various specific settings where the phrase is used. CMummert 20:14, 7 December 2006 (UTC)
replies: text: I agree with Carl here. As far as I'm aware, there is no agreed abstract definition of what a Gödel numbering is in general, mainly because the need for such a definition has never been felt. Ordinarily one doesn't treat Gödel numberings in general, so why bother to define them in general? That's not to say someone hasn't published a general definition of "Gödel numbering", but unless that definition is in common use (which it certainly isn't), it shouldn't be presented here as the default. --Trovatore 03:59, 7 January 2007 (UTC)
thread_title:  Definition 

message: It may be helpful to relate the history for the Gödel numbering concept for the readers, such as its use in Georg Cantor's transfinite numbers, in Gödel's theorems, and in Church's expositions. --Ancheta Wis 06:06, 13 January 2007 (UTC)
thread_title:  History 

message: Ok, let's use base-4 mapping: 'x' => 0, 'n' => n 1s, '+' => 2, '=' => 3
The formula F(x)=x+1 will be encoded by sequence 0, 2, 1 or number 1204 or in decimal notation= 1*42+ 2*41 + 0*40 = 16  + 8 = 2410=#F. Here #F is the number of the form F(x). The statements are derived by inserting constant numbers instead of x in the form. For instance #F(1) would be 25 and #F(2) would be 1*43+2*42+ 1*41 + 1*40. Generally, the number of the statement for constant n will be 
\#F(n)=\sum_{i=0}^{n-1}{4^i} + 2*4^n + 1*4^{n+1} .
Now, I want to get the Gödel number for the statement F(n=#F). However, the equation
\#F(\#F)=g=\sum_{i=0}^{g-1}{4^i} + 2*4^g + 1*4^{g+1} 
have no solutions in natural numbers! So, the Gödel number depends on encoding and may be irrational or not exist if encoding is wrong. What is the proper encoding? Getting the number I want to decode it and look at the statement F(#F). --Javalenok 16:22, 15 January 2007 (UTC)
replies: text: Rebecca Goldstein (2005), Incompleteness  uses base 10: pp.172-3
Basic sign	Gödel number	meaning
~         	1           	not
→         	2           	if .. then ..
x         	3           	variable
=         	4           	equals
0         	5           	zero
s         	6           	successor of
(         	7           	l-paren
)         	8           	r-paren
'         	9           	prime
--Ancheta Wis 00:07, 16 January 2007 (UTC)
thread_title:  Example 

message: I am a music theorist with an interest in logical systems. From a layperson's perspective our English page is woefully inadequate on this topic. "Goedel lite" is not very helpful even if the mapping concept is properly conveyed. For more useful articles on this topic, see the German and Chinese versions.
68.19.242.28 07:50, 25 April 2007 (UTC)
replies: text: Do you read those languages? Can you convey to other editors what might be useful to add? The essential idea of mapping concepts to counting numbers is already in the article. --Ancheta Wis 10:57, 25 April 2007 (UTC)
replies: text: Perhaps we could appeal to the relevant Wikipedia embassies to give us, perhaps not a complete translation, but some idea of what the content of the other articles is. I'm not inclined to, but it's an idea. Hmmm, German, English, Chinese, formal logic, music... an interesting combination. Andrewa 04:58, 7 September 2007 (UTC)

text: By Chinese, I assume you mean the article at zh:哥德尔数. There's also one at zh-yue:Gödel號數 in Cantonese, but it's very short. Andrewa 03:59, 8 September 2007 (UTC)
thread_title:  From a non-expert 

message: There's an article at Gödel numbering for sequences which has only one principle author, no links to other languages, no talk page yet, notes which give some sources (but nothing online unless you read Hungarian) but no other citations, and reads to me very much like original research. It may be the cause of some of the confusion here. Other comments? Andrewa 04:43, 8 September 2007 (UTC)
replies: text: Thanks for the link. My heart warmed as I read the article, which is clear, easy to read, has citations, and uses standard concepts such as the integer sequence. What came to mind immediately is a useful coding system promulgated by AT& T Research [online encyclopedia of integer sequences] which shows clearly that codes, of which a Gödel code (a redirect to this article) is an example, are arbitrary in spite of being well defined. My inclination is to let the article live in peace in the encyclopedia. We need more like it, in my opinion. --Ancheta Wis 09:02, 8 September 2007 (UTC)

text: Here is a poster for the OEIS, which has over 130,000 sequences at the moment. See the article: On-Line Encyclopedia of Integer Sequences --Ancheta Wis 09:24, 8 September 2007 (UTC)
replies: text: I'm familiar with OEIS. But what has this to do with Gödel numbering? Andrewa 03:00, 9 September 2007 (UTC)
replies: text: Only Gödel's use of natural numbers as a code, and the use of the codes of OEIS to denote sequences of natural numbers. --Ancheta Wis 10:38, 9 September 2007 (UTC)
replies: text: OK. I'm unconvinced, but I don't intend to take it any further. Mainly I just wanted some other people to have a look at the issues I raised. Glad you like the article. Andrewa 00:36, 11 September 2007 (UTC)
thread_title: Another different meaning

message: Note that K is infinite in ordinary maths. For instance, there are infinitely many
Mathieu functions.  —Preceding unsigned comment added by 87.194.34.71 (talk) 12:49, 14 July 2009 (UTC)
replies: text: The example presented there is about base-K numbers. That does not make sense unless K is finite. There are other examples that would work with an infinite number of basic symbols. For example, the one originally used by Gödel and presented in the first section. — Carl (CBM · talk) 13:20, 14 July 2009 (UTC)
thread_title:  Infinity 

message: This article is currently called Gödel number, but it seems to be completely about Gödel numbering. A Gödel number is only defined based on an overall Gödel numbering and the fact that a statement has Gödel number 258 is completely meaningless without knowing the Gödel numbering used. Any objections to moving the article? Cheers, — sligocki (talk) 02:21, 9 December 2009 (UTC)
replies: text: Alright, done. — sligocki (talk) 09:02, 16 December 2009 (UTC)
thread_title:  Move to Gödel numbering? 

message: Its very curious to postulate mappings of mathematical statements/objects to numbers, and then to not exclude the possibility of manipulations of equations/statements to correspond to functions on the numbers,... but Id love to see real practical examples of this (the wiki page is more like: hey we can encode, and in theory we can do operations on the encoding,...)
Are their explicit examples of parts of mathematics that are very well served by a canonical godel numbering which displays how certain manipulations/simplifications are reduced to simple algorithms on the godel numbers of their data?
Perhaps skolemnization, or a sensible numbering on a kind of graphs, with corresponding graph manipulations becoming arithmetical on their numbers?
Without examples its just: lets encode y'all!  — Preceding unsigned comment added by 83.134.175.183 (talk) 07:28, 21 January 2012 (UTC)
thread_title:  Examples of practical applications? 

message: 03:56, 29 June 2014 (UTC)
thread_title:  Moving the good faith contribution here. Discussion welcome 

message: Tried to follow the link to 'Visualising...' but was told that access was not authorised (university site).  I have replaced the link with one to the same article on the author's blog - though this link may be less long-lived it is accessible. However the actual article is pretty technical and only refers to Gödel numbering in passing, so maybe the link should be removed entirely - since guidelines suggest: "Links in the "External links" section should be kept to a minimum."
--Armulwp (talk) 23:15, 6 January 2017 (UTC)
thread_title:  Further Reading 

message: The German article de:Gödelnummer states, in its first sentence, the opposite of . Which is true? ◀ Sebastian 11:40, 28 April 2020 (UTC)
replies: text: Both are true: Being a function, a Gödel numbering assigns to every formula a unique natural number (cf. also the English lead). However, there are many such functions (this is meant in ). - Jochen Burghardt (talk) 13:44, 28 April 2020 (UTC)
replies: text:  Thanks. Now that you wrote this I realize that the second part of the sentence says that clearly. The problem was in the first part, which is obviously meant to state the same more concisely, but leaves an ambiguity, if taken by itself. The root of that ambiguity is in the definition of the lemma: Does the term “Gödel numbering” refer to the general method, or to one particular choice of numbering? The first sentence of the article suggests the latter, but the phrasing “A Gödel numbering” in the section in question suggests the former. This could be mitigated by leaving out the article “A”, which, according to our article, suggests “a general statement about any such thing”. ◀ Sebastian 13:14, 11 May 2020 (UTC)
replies: text:  - I hope it is better now. - Jochen Burghardt (talk) 15:01, 11 May 2020 (UTC)
thread_title:  Uniqueness – or not?