Talk:Gödel numbering

1491:, right back to a condition where selection of one element from a set of answers is a 'computation'; the difficulty is that you need a 'recognizer' to figure out how to make the selection. That's probably the reason there was no citation for the statements made in the linked articles. The statements don't stand up. There is no current theory of recognition for such a recognizer of an answer in some requested computation. If you already knew the answer, you wouldn't need to ask the question. If you didn't know the answer and the telephone switch connected to a wrong number you still would be stuck; we would still have the Knowledge conundrum -- how do you know an answer is right, unless you already know the answer? 129: 119: 98: 65: 442:

and informative than 'mathematics' while obviously implying the subject matter to be mathematics. There is a fine line between targeting a broader audience and being as accurate as possible without being overly specific. I think 'formal number theory' is a happy medium. If I were to consider a runner-up, it would be your previously proposed 'mathematical logic'. Both seem fine to me. I used 'formal number theory' because it is popular in the literature (see Kleene,

426:. The first sentence should serve as a introduction for the reader, and set the context for the rest of the page. After reading the first sentence most readers, even non-mathematicians (remember this is wikipedia and not mathworld) should have a rough idea what we are going to talk about. As I wrote in my edit summary I have no intention to start an edit war, so I will leave the article as is.I just wanted to voice my concerns. 791: 21: 56: 1952: 1936:

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

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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

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I do not know much in the way of number theory, but it appears that Godel numbering is what is needed to construct a mathematical equivalent to the usage of the phrase "This page intentionally blank" on blank pages. It is self-refuting, in that it falsifies itself by its very existence on the page in

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Formal number theory is essentially propositional logic with peano arithmetic and/or any extension thereof. The domain is the natural numbers (or (0 and) the positive integers). It is also known as simply 'arithmetic' - i.e., theories closed under addition and multiplication in which the naturals can

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The term is actually not specific at all, though it is accurate. It is not as general as mathematics, but it is far more informative. If I said, "what subject do you think formal number theory falls in?" to a non-mathematician, I'm sure they'd be able to respond "mathematics". So it is more specific

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that other people like me might appreciate a practical demo along these lines. Anome - A paragraph along these lines at the end could be a good idea. I guess it's worth pointing out the difference between a single instantiation like this, and the mathematician's quest for a general exposition.

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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

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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

1526:. The general approach is to establish a contradiction by assuming that something does exist, and then showing that this means that it both does and doesn't have a Gödel number. I fear that these other applications aren't Gödel numbers at all. Happy to be proved wrong, that means I learn something. 627:

There shouldn't be a proof of the undecidability theorem here anyway. This article is linked from several articles on computation theory, and anyone following the links is going to be horribly confused as the discussion gets into needless detail on Gödel's own application and says nothing about the

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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

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Anome is correct, there are a number (infinite?) of different mappings that are possible for use in Gödel's proof. I believe Gödel's original numbering system, however, used prime numbers taken to different powers to differentiate between variables, sentential variables and predicate variables. The

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John - you may well be right. But in defense of a separate article, I thought there could be a place for a slightly less formal article on this topic. Gödel lite, as it were. I personally struggled to grasp the technical side of the argument (as expert readers can no doubt tell(!)), and I hoped

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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:

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The shortcoming is due to the fact that any number to the power of zero is set to one, by arbitrary definition of the zero power. Logically, the number of multiplications of a number by itself is what is indicated by an integer power. If it is not multiplied at all, say x to the zero power, the

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Secondly, it is impossible in a straight line, so to speak, to encode an algorithm which in one step produces a result from a single (say) number as input and then the next step is its complete reversal, i.e. from the results of the computation to derive the original input point. The shortcoming

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GN cannot be performed in all of mathematics or computer science. It doesn't make any sense to say "In mathematics and computer science, GN is..." because you cannot perform GN in numerous branches of mathematics - i.e. you cannot perform GN in the theory of real closed fields because you cannot

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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

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For example, cyclic or self-referential statements cannot be for one identified easily as this would require solution of the entire sequence of integer numbers obtained for each statement in turn and identification of the closure of a loop (there can indeed be many loops in a logically flawed

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Perhaps we could appeal to the relevant Knowledge 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.

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Note that the exact form of the Gödel number encoding is unimportant: there just needs to be a 1:1 mapping between local statements and manipulable expressions such as integers. You can do clever stuff with arithmetic, or you can just follow Chaikin and use Lisp S-expressions --

825:=#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*4+2*4+ 1*4 + 1*4. Generally, the number of the statement for constant n will be 1392:

I fear that this would just look silly. The whole idea of Gödel numbers is fairly esoteric. You need to know quite a lot of mathematics before you'd have the slightest clue that they were useful. The things that they are used to prove seem nonsensical to the

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Formal number theory may be the correct term to use, but the term is too specific. Most people will have no idea what you are talking about (e.g. myself). So I think it is better, even if slightly incorrect, to use more generic fields like

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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).

1741:. 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, — 1780:

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?

1836:! There is though a very serious flaw with Godel numbering as it is based on integer factorization. It is impossible to encode the number zero ("0") with the basic theorem of arithmetic used in the encoding as described above. 1816:

There is though a very serious flaw with Godel numbering as it is based on integer factorization. It is impossible to encode the number zero ("0") with the basic theorem of arithmetic used in the encoding as described above.

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occurs in the all of the supposedly 'different' meanings. What was intriguing to me was the idea that a 'computation' might be replaced by an 'assignment' in the sense of a telephone switchboard operator, who would physically

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define the naturals. I am reverting again to formal number theory, and I hope I have made myself clear as to why. If not, we'll need to talk about it in the article's discussion page. I don't see why we should need to, though.

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These are just some fundamental direct flaws with the so called Incompleteness Theorems. The other serious flaw is the consideration of "closure" or "independence" of the axioms comprising and axiomatic space to begin with.

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275:, which is where you probably got your info. from in the first place. I actually have Godel's original paper onhand but have never really read more than the introduction. It uses a lot of outdated terminology and symbolism. 1500:

analogy, a telephone number is a code for a person. Dial the number and talk to the person. We do not yet have a technology to connect remotely with another person without that code. There are technologies such as the

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I agree--the sketch here doesn't seem correct. I don't believe you can construct the required sentence using the R defined here--you need a statement form that describes self-unprovability, not just provability. The

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I speak no German to mention, but even I can tell it's not a perfect article in German. But food for thought, and they do have the advantage that much of Gödel's work on this was originally published in German.

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A Gödel numbering can be interpreted as an encoding in which a number is assigned to each symbol of a mathematical notation, after which a sequence of natural numbers can then represent a sequence of strings

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My advice is, be aware that many who do come here do not have the understanding needed to make sense of what they read, but will acquire it, and among those who struggle hardest at this may be the next

1587:(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. -- 1227:

Can you give the Gödel number G=#F(#F) for F(x)=~x? The tells that the Gödel number for the Gödel sentence is 'precisely defined'. How do they derive it if the provability predicate P is not known? --

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mapping? Though injectiveness is implied by the fact that reference is made to the mapping's inverse (which doesn't exist for non-injective maps), I think injectiveness should be stated explicitly.

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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. --

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Anyway, the numbering method used in this wiki article is capable of coding sequences of formulae, so I'm not sure why you thought otherwise. Why did you? Because it isn't explicitly stated?

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Godel's original used only 7 elementary operators and assigned primes, squared primes, and cubed primes to the 3 variables u listed respectively above - at least according to E. Nagel in

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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

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where the name of a 'person' connects you right to the 'person' without the code, but the idea has been around for over 30 years with no commercial product in sight, for the moment. --

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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

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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." --

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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. --

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is also short, but it does look in some (not all) ways a better article than ours. Ours rambles and speculates. The German article, by comparison, has only the following sections:

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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. --

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Why not add a simple example or two near the beginning. Assign numbers to symbols as in one of the comments below and do a formula-to-number and a number-to formula example.

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mapping currently demonstrated in the article only works for encoding symbols and formulas but not sequences of formulas, a necessary characteristic for Gödel's proof. --

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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.

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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?

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I'm afraid I don't think any of that has much to do with Gödel numbering. The only use I know of these numbers is to establish that things such as solutions to the

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are fascinating to the uninitiated, so we do need to do our best. But we musn't stray into inaccuracy for the sake of simplicity. It's not a simple topic.

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is a confusing one, particularly to non-specialists. Even many specialists are still trying to figure out exactly what the consequences are, both for

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Perhaps skolemnization, or a sensible numbering on a kind of graphs, with corresponding graph manipulations becoming arithmetical on their numbers?

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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

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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

1583:. What came to mind immediately is a useful coding system promulgated by AT& T Research which shows clearly that codes, of which a 1416: 1687: 830: 313: 248: 216: 1863: 1097: 142: 103: 1844:

arises by the use of the fundamental theorem of arithmetic, i.e. prime factorization, in that it does not contain the zero value.

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arises by the use of the fundamental theorem of arithmetic, i.e. prime factorization, in that it does not contain the zero value.

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The current definition (without the vandalism) is nonsense. Does anybody have a definition that satisfies these characteristics?

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that undergirded their most famous work. So for the sake of these strugglers, I commend all those struggling with this article.

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which give some sources (but nothing online unless you read Hungarian) but no other citations, and reads to me very much like

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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

40:. Please limit discussion to improvement of this article. You may wish to ask factual questions about Gödel numbers at the 1634:

Only Gödel's use of natural numbers as a code, and the use of the codes of OEIS to denote sequences of natural numbers. --

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There is a very different meaning of the term Gödel numbering relating to numberings of the class of computable functions.

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Should this article more properly be moved to "Gödel number", or are the numbers better known by the "phonetic" form?

1964: 1926: 1502: 362:? I am not sure what fields of study to use in the introduction. I know Gödel numbers are used in the proof of the 1381: 712:

without comment here. Either someone agrees with me, or they feel that the confusing material has been removed.

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at the start of the article because I thought you deleted my previous edit because it was to specific. What is

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x, ¬R(v,x) Can also be interpeted to mean "The negation of a proposition of type v can be proved", yielding:

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section and replace it with a discussion of the various specific settings where the phrase is used.

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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that the final sequence of numbers represents. So it is a little confusing… Included in Godel’s

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is infinite in ordinary maths. For instance, there are infinitely many Mathieu functions.

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I think (if this is true) that we should find this meaning and include it in the article.

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Now, I want to get the Gödel number for the statement F(n=#F). However, the equation

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which has only one principle author, no links to other languages, no talk page yet,

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to the main article as John suggests. I'll leave that decision to the meta-mind.

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don't exist. They have no practical application, only this theoretical use in

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Please, if you tag an article, give us some idea why you are tagging it.

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3 The formula F(x)=x+1 will be encoded by sequence 0, 2, 1 or number 120

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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

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I'm familiar with OEIS. But what has this to do with Gödel numbering?

1566:. It may be the cause of some of the confusion here. Other comments? 1488: 1466:

I followed the links you highlighted and discovered that the keyword

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can encode, and in theory we can do operations on the encoding,...)

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More general than just formulas of an effective first-order language

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Based on this I am commenting out said statement in the article. --

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where telephone operators were replaced by computerized telephone

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in a way that would make sense to someone who doesn't know what a

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I'm not trying to be elitist here. But you wouldn't try to define

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I'm trying to figure out why certain pages aren't rendering a TOC

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is. I feel that some of the edits to this article, while I hope

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just because a statement says something doesn't make it true).

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for general discussion about Gödel numbers. Any such comments

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The thing about the DA before GN was a mistake. I meant the

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Moving the good faith contribution here. Discussion welcome

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connection between numberings and programming languages.

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or something like that - that's what should be there)

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or in decimal notation= 1*4+ 2*4 + 0*4 = 16 + 8 = 24

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notice is helpful, nor that this article belongs in

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That does not make sense unless 1602:On-Line Encyclopedia of Integer Sequences 1055: 1036: 1016: 1011: 999: 988: 955: 923: 904: 884: 879: 867: 856: 832: 578:Shouldn't Gödel mapping be defined as an 519: 499: 467: 813:Ok, let's use base-4 mapping: 'x' =: --> 687:What are the specific ways in which the 1998:Knowledge vital articles in Mathematics 370:. So I think it is perhaps best to use 94: 53: 1737:, but it seems to be completely about 2013:C-Class vital articles in Mathematics 406:GN." vs. "...is called GN.". Cheers, 7: 546:Requesting equivalent statement for 140:This article is within the scope of 1848:result should be zero and not one. 1829:result should be zero and not one. 1772:Examples of practical applications? 1340:Gödel numbering of Turing machines 1285:, I assume you mean the article at 494:(Still flawed, but have to include 83:It is of interest to the following 2023:High-priority mathematics articles 966: 957: 834: 521: 501: 469: 14: 1733:This article is currently called 650:which is the result of this tag. 589:Informal proof doesn't make sense 160:Knowledge:WikiProject Mathematics 1993:Knowledge level-5 vital articles 1950: 306:are sequences of formulas too. 163:Template:WikiProject Mathematics 127: 117: 96: 63: 54: 19: 1417:Gödel's incompleteness theorems 1325:(which Babelfish translates as 180:This article has been rated as 2003:C-Class level-5 vital articles 1660:00:36, 11 September 2007 (UTC) 972: 963: 846: 840: 633:08:24, 27 September 2006 (UTC) 616:Godel's incompleteness theorem 249:Gödel's incompleteness theorem 217:Gödel's incompleteness theorem 1: 1959:- I hope it is better now. - 1856:Moved the contribution here. 1766:09:02, 16 December 2009 (UTC) 1639:10:38, 9 September 2007 (UTC) 1622:03:00, 9 September 2007 (UTC) 1609:09:24, 8 September 2007 (UTC) 1592:09:02, 8 September 2007 (UTC) 1571:04:43, 8 September 2007 (UTC) 1556:Gödel numbering for sequences 1540:03:02, 8 September 2007 (UTC) 1531:02:44, 8 September 2007 (UTC) 1510:09:46, 7 September 2007 (UTC) 1485:history of computing hardware 1461:04:14, 7 September 2007 (UTC) 1442:04:47, 7 September 2007 (UTC) 1386:18:51, 6 September 2007 (UTC) 1358:04:33, 8 September 2007 (UTC) 1302:03:59, 8 September 2007 (UTC) 1273:04:58, 7 September 2007 (UTC) 720:04:28, 7 September 2007 (UTC) 623:20:21, 2 September 2006 (UTC) 541:04:47, 12 November 2005 (UTC) 514:, shiftleft, shiftright, and 492:04:29, 12 November 2005 (UTC) 336:the DA before GN is idiomatic 245:Gödel numbers - demonstration 154:and see a list of open tasks. 2018:C-Class mathematics articles 1803:07:28, 21 January 2012 (UTC) 1751:02:21, 9 December 2009 (UTC) 1323:zahlentheoretischen Aussagen 1232:10:21, 16 January 2007 (UTC) 1223:00:07, 16 January 2007 (UTC) 1085:16:22, 15 January 2007 (UTC) 782:06:06, 13 January 2007 (UTC) 752:20:14, 7 December 2006 (UTC) 704:20:33, 24 October 2006 (UTC) 376:theoretical computer science 1890:23:15, 6 January 2017 (UTC) 767:03:59, 7 January 2007 (UTC) 642:I'm not convinced that the 251:article, while redirecting 2039: 1931:13:44, 28 April 2020 (UTC) 1912:11:40, 28 April 2020 (UTC) 1840:construct of statements). 1821:construct of statements). 1813:03:56, 29 June 2014 (UTC) 1503:content-addressable memory 1327:pay-theoretical statements 1259:10:57, 25 April 2007 (UTC) 1249:07:50, 25 April 2007 (UTC) 1090:Rebecca Goldstein (2005), 794:, presaged the concept. -- 695:that the article was then 604:22:18, 25 April 2006 (UTC) 588: 247:, and link to it from the 1867:03:54, 29 June 2014 (UTC) 1723:13:20, 14 July 2009 (UTC) 1696:12:49, 14 July 2009 (UTC) 1550:Another different meaning 804:11:32, 29 June 2011 (UTC) 382:14:52, 11 Nov 2004 (UTC) 267:18:13, 24 Oct 2004 (UTC) 179: 112: 91: 1969:15:01, 11 May 2020 (UTC) 1945:13:14, 11 May 2020 (UTC) 1729:Move to Gödel numbering? 1410:, are equally misguided. 548:Intentionally blank page 527:{\displaystyle \forall } 475:{\displaystyle \forall } 450:07:36, 12 Nov 2004 (UTC) 430:06:01, 12 Nov 2004 (UTC) 410:05:19, 12 Nov 2004 (UTC) 322:22:29, 21 May 2012 (UTC) 259:08:33 16 Jul 2003 (UTC) 240:06:24 16 Jul 2003 (UTC) 232:23:48 15 Jul 2003 (UTC) 223:11:15 Apr 9, 2003 (UTC) 186:project's priority scale 1448:Very different meaning? 1297:, but it's very short. 788:Christine Ladd-Franklin 507:{\displaystyle \infty } 284:08:17, 4 Nov 2004 (UTC) 143:WikiProject Mathematics 1988:C-Class vital articles 1554:There's an article at 1496:To follow-up with the 1452:Current article reads 1307:The German article at 1289:. There's also one at 1100:uses base 10: pp.172-3 1071: 1010: 939: 878: 528: 508: 476: 1363:Add Simple Example(s) 1329:but I think may mean 1072: 984: 940: 852: 529: 509: 477: 364:incompleteness result 70:level-5 vital article 1895:Uniqueness – or not? 1335:number theory proofs 1331:well-formed formulas 954: 831: 618:is more accurate. -- 518: 498: 466: 368:computability theory 360:formal number theory 166:mathematics articles 1907:. Which is true? ◀ 1899:The German article 1871:Discussion welcome 1598:poster for the OEIS 1321:Gödel numbering of 743:I would remove the 1919:Lack_of_uniqueness 1905:Lack_of_uniqueness 1067: 935: 524: 504: 472: 444:Mathematical Logic 400:indefinite article 372:mathematical logic 135:Mathematics portal 79:content assessment 31: 1793:comment added by 1756:Alright, done. — 1721: 1686:comment added by 1564:original research 1388: 1376:comment added by 1315:Formal definition 1237:From a non-expert 1216: 1215: 567: 566:• 2005-12-4 17:47 312:comment added by 304:class of formulas 215:just redirect to 200: 199: 196: 195: 192: 191: 48: 47: 27: 2030: 1961:Jochen Burghardt 1958: 1954: 1953: 1923:Jochen Burghardt 1805: 1711: 1698: 1415:The problem is, 1408:well-intentioned 1371: 1104: 1103: 1076: 1074: 1073: 1068: 1066: 1065: 1041: 1040: 1022: 1021: 1020: 1009: 998: 944: 942: 941: 936: 934: 933: 909: 908: 890: 889: 888: 877: 866: 815:n 1s, '+' =: --> 564: 555: 533: 531: 530: 525: 513: 511: 510: 505: 481: 479: 478: 473: 424:computer science 352:computer science 341:I have put the 324: 168: 167: 164: 161: 158: 137: 132: 131: 121: 114: 113: 108: 100: 93: 76: 67: 66: 59: 58: 50: 23: 22: 16: 2038: 2037: 2033: 2032: 2031: 2029: 2028: 2027: 1978: 1977: 1951: 1949: 1897: 1877: 1875:Further Reading 1811: 1788: 1774: 1739:Gödel numbering 1731: 1681: 1674: 1552: 1520:halting problem 1450: 1435:tensor calculus 1400:just intonation 1378:207.195.244.207 1365: 1239: 1051: 1032: 1012: 952: 951: 919: 900: 880: 829: 828: 824: 820: 811: 775: 727: 675:Turing machines 640: 591: 576: 562: 551: 516: 515: 496: 495: 486: 464: 463: 461: 387:be constructed. 332: 329:Gödel numbering 307: 265:128.253.167.158 205: 165: 162: 159: 156: 155: 133: 126: 106: 77:on Knowledge's 74: 64: 20: 12: 11: 5: 2036: 2034: 2026: 2025: 2020: 2015: 2010: 2005: 2000: 1995: 1990: 1980: 1979: 1976: 1975: 1974: 1973: 1972: 1971: 1901:de:Gödelnummer 1896: 1893: 1876: 1873: 1855: 1810: 1807: 1795:83.134.175.183 1773: 1770: 1769: 1768: 1730: 1727: 1726: 1725: 1673: 1670: 1669: 1668: 1667: 1666: 1665: 1664: 1663: 1662: 1646: 1645: 1644: 1643: 1642: 1641: 1627: 1626: 1625: 1624: 1612: 1611: 1594: 1551: 1548: 1547: 1546: 1545: 1544: 1543: 1542: 1524:indirect proof 1513: 1512: 1493: 1492: 1449: 1446: 1445: 1444: 1421: 1420: 1412: 1411: 1395: 1394: 1364: 1361: 1350: 1349: 1346: 1345: 1344: 1338: 1319: 1316: 1309:de:Gödelnummer 1305: 1304: 1291:zh-yue:Gödel號數 1278: 1277: 1276: 1275: 1262: 1261: 1238: 1235: 1214: 1213: 1210: 1207: 1203: 1202: 1199: 1196: 1192: 1191: 1188: 1185: 1181: 1180: 1177: 1174: 1170: 1169: 1166: 1163: 1159: 1158: 1155: 1152: 1148: 1147: 1144: 1141: 1137: 1136: 1135:if .. then .. 1133: 1130: 1126: 1125: 1122: 1119: 1115: 1114: 1111: 1108: 1102: 1101: 1092:Incompleteness 1064: 1061: 1058: 1054: 1050: 1047: 1044: 1039: 1035: 1031: 1028: 1025: 1019: 1015: 1008: 1005: 1002: 997: 994: 991: 987: 983: 980: 977: 974: 971: 968: 965: 962: 959: 932: 929: 926: 922: 918: 915: 912: 907: 903: 899: 896: 893: 887: 883: 876: 873: 870: 865: 862: 859: 855: 851: 848: 845: 842: 839: 836: 822: 818: 810: 807: 774: 771: 770: 769: 741: 740: 737: 734: 726: 723: 653:Certainly the 639: 636: 590: 587: 585: 575: 572: 570: 550: 544: 523: 503: 484: 471: 460: 457: 456: 455: 454: 453: 452: 451: 434: 433: 432: 431: 412: 411: 395: 394: 389: 388: 356: 355: 331: 326: 296: 295: 288: 286: 285: 277: 276: 226: 210: 204: 201: 198: 197: 194: 193: 190: 189: 178: 172: 171: 169: 152:the discussion 139: 138: 122: 110: 109: 101: 89: 88: 82: 60: 46: 45: 42:Reference desk 34:may be removed 24: 13: 10: 9: 6: 4: 3: 2: 2035: 2024: 2021: 2019: 2016: 2014: 2011: 2009: 2006: 2004: 2001: 1999: 1996: 1994: 1991: 1989: 1986: 1985: 1983: 1970: 1966: 1962: 1957: 1948: 1947: 1946: 1943: 1939: 1934: 1933: 1932: 1928: 1924: 1920: 1916: 1915: 1914: 1913: 1910: 1906: 1902: 1894: 1892: 1891: 1887: 1883: 1874: 1872: 1869: 1868: 1865: 1862: 1859: 1853: 1849: 1845: 1841: 1837: 1834: 1830: 1826: 1822: 1818: 1814: 1808: 1806: 1804: 1800: 1796: 1792: 1785: 1782: 1778: 1771: 1767: 1763: 1759: 1755: 1754: 1753: 1752: 1748: 1744: 1740: 1736: 1728: 1724: 1719: 1715: 1709: 1705: 1701: 1700: 1699: 1697: 1693: 1689: 1685: 1679: 1671: 1661: 1658: 1654: 1653: 1652: 1651: 1650: 1649: 1648: 1647: 1640: 1637: 1633: 1632: 1631: 1630: 1629: 1628: 1623: 1620: 1616: 1615: 1614: 1613: 1610: 1607: 1603: 1599: 1595: 1593: 1590: 1586: 1582: 1579: 1575: 1574: 1573: 1572: 1569: 1565: 1561: 1557: 1549: 1541: 1538: 1534: 1533: 1532: 1529: 1525: 1521: 1517: 1516: 1515: 1514: 1511: 1508: 1504: 1499: 1495: 1494: 1490: 1486: 1482: 1478: 1475:a conductive 1474: 1469: 1465: 1464: 1463: 1462: 1459: 1455: 1447: 1443: 1440: 1436: 1432: 1428: 1423: 1422: 1418: 1414: 1413: 1409: 1405: 1404:musical scale 1401: 1397: 1396: 1391: 1390: 1389: 1387: 1383: 1379: 1375: 1368: 1362: 1360: 1359: 1356: 1347: 1342: 1341: 1339: 1336: 1332: 1328: 1324: 1320: 1317: 1314: 1313: 1312: 1310: 1303: 1300: 1296: 1292: 1288: 1284: 1280: 1279: 1274: 1271: 1266: 1265: 1264: 1263: 1260: 1257: 1253: 1252: 1251: 1250: 1247: 1243: 1236: 1234: 1233: 1230: 1225: 1224: 1221: 1211: 1208: 1205: 1204: 1200: 1197: 1194: 1193: 1189: 1186: 1183: 1182: 1179:successor of 1178: 1175: 1172: 1171: 1167: 1164: 1161: 1160: 1156: 1153: 1150: 1149: 1145: 1142: 1139: 1138: 1134: 1131: 1128: 1127: 1123: 1120: 1117: 1116: 1112: 1110:Gödel number 1109: 1106: 1105: 1099: 1098:0-393-05169-2 1096: 1093: 1089: 1088: 1087: 1086: 1083: 1077: 1062: 1059: 1056: 1052: 1048: 1045: 1042: 1037: 1033: 1029: 1026: 1023: 1017: 1013: 1006: 1003: 1000: 995: 992: 989: 985: 981: 978: 975: 969: 960: 949: 946: 930: 927: 924: 920: 916: 913: 910: 905: 901: 897: 894: 891: 885: 881: 874: 871: 868: 863: 860: 857: 853: 849: 843: 837: 826: 816:2, '=' =: --> 814:0, 'n' =: --> 808: 806: 805: 801: 797: 793: 789: 784: 783: 780: 772: 768: 765: 761: 756: 755: 754: 753: 750: 746: 739:Used in print 738: 735: 732: 731: 730: 724: 722: 721: 718: 713: 711: 706: 705: 702: 698: 694: 690: 685: 683: 678: 676: 672: 668: 664: 660: 656: 651: 649: 645: 644:{{confusing}} 637: 635: 634: 631: 625: 624: 621: 617: 613: 607: 605: 602: 598: 594: 586: 583: 581: 574:Injective map 573: 571: 568: 565: 560: 559: 549: 545: 543: 542: 539: 534: 493: 490: 483: 458: 449: 445: 440: 439: 438: 437: 436: 435: 429: 425: 421: 416: 415: 414: 413: 409: 405: 401: 397: 396: 391: 390: 385: 384: 383: 381: 377: 373: 369: 365: 361: 354: 353: 349: 344: 343: 342: 339: 337: 330: 327: 325: 323: 319: 315: 311: 305: 301: 292: 291: 290: 283: 279: 278: 274: 273:Gödel's Proof 270: 269: 268: 266: 260: 258: 254: 250: 246: 241: 239: 233: 231: 224: 222: 218: 214: 213:Goedel number 208: 202: 187: 183: 182:High-priority 177: 174: 173: 170: 153: 149: 145: 144: 136: 130: 125: 123: 120: 116: 115: 111: 107:High‑priority 105: 102: 99: 95: 90: 86: 80: 72: 71: 61: 57: 52: 51: 43: 39: 35: 30: 26:This page is 25: 18: 17: 1955: 1898: 1878: 1870: 1854: 1850: 1846: 1842: 1838: 1835: 1831: 1827: 1823: 1819: 1815: 1812: 1789:— Preceding 1786: 1783: 1779: 1775: 1738: 1735:Gödel number 1734: 1732: 1707: 1703: 1688:87.194.34.71 1677: 1675: 1559: 1553: 1480: 1476: 1472: 1467: 1453: 1451: 1393:uninitiated. 1369: 1366: 1351: 1334: 1330: 1326: 1322: 1306: 1282: 1246:68.19.242.28 1244: 1240: 1226: 1217: 1091: 1078: 950: 947: 827: 812: 785: 776: 759: 744: 742: 728: 714: 708:The tag was 707: 696: 692: 688: 686: 681: 679: 663:Formal logic 654: 652: 643: 641: 626: 612:proof sketch 608: 601:82.41.211.70 599: 595: 592: 584: 579: 577: 569: 557: 554:question. — 552: 535: 487: 462: 443: 403: 399: 357: 345: 340: 335: 333: 314:75.82.51.192 308:— Preceding 303: 299: 297: 287: 272: 261: 253:Gödel number 244: 242: 234: 225: 209: 206: 181: 141: 85:WikiProjects 68: 1938:our article 1864:| contribs) 1858:Ancheta Wis 1682:—Preceding 1636:Ancheta Wis 1606:Ancheta Wis 1589:Ancheta Wis 1537:Ancheta Wis 1507:Ancheta Wis 1372:—Preceding 1256:Ancheta Wis 1220:Ancheta Wis 1107:Basic sign 796:Ancheta Wis 779:Ancheta Wis 671:Alan Turing 659:Mathematics 420:mathematics 348:mathematics 334:What does 157:Mathematics 148:mathematics 104:Mathematics 29:not a forum 1982:Categories 1676:Note that 1596:Here is a 1585:Gödel code 1468:assignment 1431:Heisenberg 1242:versions. 786:Note that 745:Definition 725:Definition 697:inaccurate 667:urban myth 538:Hackwrench 489:Hackwrench 428:MathMartin 380:MathMartin 221:John Owens 38:refactored 1942:Sebastian 1909:Sebastian 1370:~reader 1295:Cantonese 1229:Javalenok 1146:variable 1082:Javalenok 764:Trovatore 638:Confusing 580:injective 448:Nortexoid 446:, 1967). 408:Nortexoid 282:Nortexoid 230:The Anome 73:is rated 1791:unsigned 1758:sligocki 1743:sligocki 1684:unsigned 1672:Infinity 1581:sequence 1498:encoding 1489:switches 1427:Einstein 1374:unsigned 1348:see also 1201:r-paren 1190:l-paren 1113:meaning 749:CMummert 693:evidence 661:and for 630:Gazpacho 459:Negation 310:unsigned 1882:Armulwp 1657:Andrewa 1619:Andrewa 1578:integer 1568:Andrewa 1528:Andrewa 1479:into a 1473:connect 1458:Andrewa 1439:Andrewa 1355:Andrewa 1343:Example 1318:Example 1299:Andrewa 1287:zh:哥德尔数 1283:Chinese 1270:Andrewa 1157:equals 809:Example 773:History 760:someone 758:to say 733:Correct 717:Andrewa 710:removed 701:Andrewa 689:article 682:article 655:subject 366:and in 338:mean ? 300:symbols 184:on the 75:C-class 1481:socket 1212:prime 620:Rictus 81:scale. 1921:). - 1861:(talk 1560:notes 1168:zero 790:, in 606:A.W. 558:BRIAN 257:Wikid 238:Wikid 219:? -- 203:Moved 62:This 1965:talk 1956:Done 1927:talk 1886:talk 1799:talk 1762:talk 1747:talk 1718:talk 1692:talk 1477:plug 1382:talk 1124:not 1095:ISBN 800:talk 792:1881 677:. 673:and 563:0918 422:and 374:and 350:and 318:talk 176:High 1714:CBM 1429:or 1333:or 1293:in 1281:By 614:in 346:In 36:or 1984:: 1967:) 1929:) 1888:) 1801:) 1764:) 1749:) 1716:· 1694:) 1604:-- 1384:) 1218:-- 1209:9 1206:' 1198:8 1195:) 1187:7 1184:( 1176:6 1173:s 1165:5 1162:0 1154:4 1151:= 1143:3 1140:x 1132:2 1129:→ 1121:1 1118:~ 1080:-- 1049:∗ 1030:∗ 1004:− 986:∑ 967:# 958:# 945:. 917:∗ 898:∗ 872:− 854:∑ 835:# 823:10 802:) 699:. 684:. 522:∀ 502:∞ 470:∀ 320:) 1963:( 1925:( 1884:( 1797:( 1760:( 1745:( 1720:) 1712:( 1708:K 1704:K 1690:( 1678:K 1380:( 1063:1 1060:+ 1057:g 1053:4 1046:1 1043:+ 1038:g 1034:4 1027:2 1024:+ 1018:i 1014:4 1007:1 1001:g 996:0 993:= 990:i 982:= 979:g 976:= 973:) 970:F 964:( 961:F 931:1 928:+ 925:n 921:4 914:1 911:+ 906:n 902:4 895:2 892:+ 886:i 882:4 875:1 869:n 864:0 861:= 858:i 850:= 847:) 844:n 841:( 838:F 819:4 798:( 404:a 316:( 294:. 188:. 87:: 44:.

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