Talk:Gödel's incompleteness theorems

Please place discussions on the underlying mathematical issues on the Arguments page. Non-editorial comments on this talk page may be removed by other editors.

Paragraph

Is the following paragraph in "first incompleteness theorem" helpful to the reading public or just confusing/not adding much??:

"Moreover, as Raatikainen (2015) states, "in favourable circumstances, it can be shown that GF is true, provided that F is indeed consistent. ... For this reason, the Gödel sentence is often called 'true but unprovable'." The word "true" is used disquotationally here: the Gödel sentence is true in this sense because it "asserts its own unprovability and it is indeed unprovable" (Smoryński 1977 p. 825; also see Franzén 2004 pp. 28–33). It is also possible to read "GF is true" in the formal sense that primitive recursive arithmetic proves the implication Con(F)→GF, where Con(F) is a canonical sentence asserting the consistency of F (Smoryński 1977 p. 840, Kikuchi and Tanaka 1994 p. 403). However, the Gödel sentence of a consistent theory may be false in some nonstandard models of arithmetic."

As is I think the section would read better with this paragraph simply deleted..68.48.241.158 (talk) 19:30, 9 July 2016 (UTC)

Honestly, I'm not hugely fond of it either.

This is the situation. The Goedel sentence of a consistent theory is true. Period. No weaseling, no waffling. Ideally, that's what we should say. We can definitely source it.

Unfortunately, the matter gets discussed to death, even though the "other side" has no valid case. This was CBM's attempt to put in something with absolutely bulletproof sourceability. --Trovatore (talk) 19:45, 9 July 2016 (UTC)

it seems unecessarily complicating..the following paragraph seems to basically cover the same ground in a way that seems more useful/appropriate for a Wikipedia article...but that paragraph seems a brief but jarring (and a bit vague) reference to more advanced technicalities and just seems distracting as stands..68.48.241.158 (talk) 20:37, 9 July 2016 (UTC)

You're kinda right, but the following paragraph buries the lede even worse than the one we're talking about. I would prefer a much less equivocal, much more direct statement, at the top of the section, pointing out that the Goedel sentence is true (full stop) but unprovable (in the theory being considered). We used to address the quibbles in an explanatory footnote, and I think that would be the way to go again. --Trovatore (talk) 21:11, 9 July 2016 (UTC)

Trovatore and I both agree that the article, like most general references, should address the truth of the Goedel sentence. Much of what was once in the explanatory footnote is now in that paragraph. It moved because we switched to a direct quote for the statement of the first incompleteness theorem. I do think that, for this kind of issue, having very clear sourcing is a must. — Carl (CBM · talk) 23:14, 9 July 2016 (UTC)

That does make sense, but I would still prefer to put the TL;DR stuff in an explanatory footnote. I'm afraid the current text really does bury the lede. --Trovatore (talk) 23:19, 9 July 2016 (UTC)

The first sentence says: in favorable circumstances, it can be shown that the Goedel sentence is true. The same sentence says: "true but unprovable". How much more direct can it be? At the same time, I don't think that the sentence about what "true" means is "TL:DR" - it's vital to understanding what it means to say that the sentence is true, which is an issue that too many people keep worrying about on this talk page. If we just say "true", we'll just get more complaints about what that is supposed to mean. — Carl (CBM · talk) 00:48, 10 July 2016 (UTC)

I think we should say flatly that it is true (when the theory in question is consistent), and deal with quibbles in a footnote. Yes, we'll get complaints, from people who have been confused by the popularizers. That should not keep us from making a direct statement. --Trovatore (talk) 00:55, 10 July 2016 (UTC)

I don't think that the meaning of "true" is really a quibble, in this case. I also think it's important to include the fact that "true" can also be read as "provable in a weak metatheory", which is to say to include the fact that the Goedel sentence is provable in a very weak theory (like PRA) from the consistency assumption. So I don't see that as a "quibble". The article needs to remain neutral between the Platonistic reading you're suggesting and a more formalistic reading of "true". — Carl (CBM · talk) 01:00, 10 July 2016 (UTC)

As you know, it is standard in mathematics to use Platonistic language whether you're a Platonist or not. I think that is the approach we should take, for the first statement we make about this. Then we can go into detail about what it means to various workers, and I think best in a footnote. --Trovatore (talk) 01:06, 10 July 2016 (UTC)

But we do just say "true" first and then explain what it means, from my perspective reading the paragraph. And the first explanation given is the Platonist one. I'm not thrilled about using a footnote, which was just there before because it was attached to the actual statement of the first theorem. If we can put the text into a regular paragraph, rather than a footnote, it's easier to read, and easier to expand on. I'm also not thrilled about the citation to the SEP article - perhaps you can find a better source for the truth of the statement? I have found it somewhat difficult to find an ideal direct quote, which is why I compromised on the SEP article. — Carl (CBM · talk) 01:09, 10 July 2016 (UTC)

At least I would get rid of "in favorable circumstances". (And also "can be shown to be".)

I generally think highly of explanatory footnotes for technical details. I think we should use them more. This strikes me as a case where one would be highly effective, as gives the main message directly at the top level, and expounds on it in a place that doesn't interrupt the flow so much.

As for direct quotes, I understand why you want to use them in contentious passages, but I think they're less than ideal in encyclopedic writing. --Trovatore (talk) 01:13, 10 July 2016 (UTC)

I agree that direct quotes are usually less than ideal, but I think this is the particular kind of circumstance where they are valuable - when there is a particular sentence which, as long as there is not a direct quote, is going to be constantly bombarded with complaints. We had the same problem with the statement of the first theorem. As for the meaning of truth here, I don't see it as a minor technical detail, but as a key part of any paragraph we would include about the truth of the Goedel sentence. In any case, if you have a better source for that paragraph, or a suggested rewording, try it out, and we can move forward from there. I spent a while looking for better sources, but what I found was not as good as what is already there. — Carl (CBM · talk) 01:19, 10 July 2016 (UTC)

paraphrasing just reads so much better...the 'background' section reads a lot better than this section, for example...the info being conveyed in this paragraph (which is correct as far as I can tell) could be better and more simply and more directly delivered using paraphrasing imo..perhaps I'll try my hand at trying something via paraphrasing...can't do it right this minute though..68.48.241.158 (talk) 01:34, 10 July 2016 (UTC)

And I agree "the favorable circumstances" clause is just so confusing..it's just a really odd way of phrasing what it's referring to, I think. It's basically this though that should be explained:

G is a well-formed proposition that indirectly points to another proposition and asserts its unprovability/undecidability within the system. But an analysis reveals that the proposition it refers to just so happens to be itself. The system, therefore, if taken to be consistent (inconsistent systems are trivial) can not decide all well-formed propositions and is necessarily incomplete. But this is exactly what the proposition asserts so the proposition is in fact true but unprovable/undecidable from within the system. Indeed, the truth of the proposition may only be arrived at via a meta-analysis from outside the system. Because of this, G is often said to be "true but unprovable."

^something more straightforward and simplified along these lines would be better/more appropriate content for the article imo..68.48.241.158 (talk) 12:40, 10 July 2016 (UTC)

Just to check, did you read the SEP article, or at least that paragraph within it, to see the reason the clause was in the original source? We could abbreviate the quote here, if we didn't want to worry about that issue. — Carl (CBM · talk) 15:41, 10 July 2016 (UTC)

just looked at it now...Wikipedia should probably be careful to not borrow too heavily from that source in its phrasing etc..I like how it explains that G isn't some universal, single proposition...I don't understand what "favorable circumstances" really means though but to simply mean the circumstances the theorems need to be applicable..68.48.241.158 (talk) 15:59, 10 July 2016 (UTC)

that is, I'm a bit lost by some of the technical detail he briefly goes into there...my intuition, however, is that it's a bit superfluous or at least distracting from what this Wikipedia article should be trying to get across in this particular passage...68.48.241.158 (talk) 16:37, 10 July 2016 (UTC)

I do not have expertise or great abilities in the technical proof itself and its technical language but I'm confident I have a very good general understanding of the proof and what it says/means...imo I also have a very good BS detector in this topic even in relation to people spouting technical BS..68.48.241.158 (talk) 16:59, 10 July 2016 (UTC)

68, you have some good stuff in there, but I would have two major objections. First, it still buries the lede. The important point is that systems of the sort under discussion cannot perfectly distinguish truth from falsity in arithmetic, because either there will be a true statement of arithmetic that the system will not find (that's the consistent case), or there will be a false statement that it will count as true (in the inconsistent case).

Relatedly, we need to point out that the Goedel sentence is a statement about the natural numbers. It is not self-referential, though it codes up certain aspects of self-reference. You do a good job of explaining at a high level how that coding part works, but it is still possible to read your text and not realize that the statement is making a very simple (in logical form, not in length complexity) statement about the naturals. --Trovatore (talk) 21:49, 10 July 2016 (UTC)

Yes, the article could potentially benefit from a stand alone section titled "truth vs provability" or some such as this trips many people up and is of particular interest to the general reader...what you refer to about the proposition is addressed in the final paragraph of the section under discussion...it's possible this paragraph should be moved to earlier in the section or the info contained within it be combined into another paragraph etc..68.48.241.158 (talk) 22:23, 10 July 2016 (UTC)

"However, the Gödel sentence of a consistent theory may be false in some nonstandard models of arithmetic." Does this sentence need to be where it is?...seems a distracting technical tangent..68.48.241.158 (talk) 00:38, 12 July 2016 (UTC)

It's quite common for people to think of truth as provability, i.e. truth in all models. So, in an elementary article, it's important to point out that the Goedel sentence is true, but it is not true in all models. — Carl (CBM · talk) 11:14, 12 July 2016 (UTC)

I also believe that the inclusion of the info about PRA is useful, along with the detailed references. These are not "tangents"; they are key aspects of the situation. — Carl (CBM · talk) 12:24, 12 July 2016 (UTC)

If this info is deemed relevant at this point in the article (instead of perhaps later in a more technical section) I think it should come in its own paragraph directly after the paragraph I just constructed...it's mostly a style thing...it just reads really clunky the way it's constructed now imo...68.48.241.158 (talk) 12:36, 12 July 2016 (UTC)

Also, can you point me to a source about "However, the Gödel sentence of a consistent theory may be false in some nonstandard models of arithmetic." It seems an odd statement to me...what are these "nonstandard models" and are they applicable/relevant to the incompleteness theorems in any direct way? What does it mean to say it is false? and how is it relevant to this section of this Wikipedia article? 68.48.241.158 (talk) 15:23, 12 July 2016 (UTC)

I'm not an expert, but the point is that Goedel sentence of a consistent theory is formally undecidable in that theory; that means that adding either the sentence or its negation (but of course not both) will yield consistent theories, and each of these will have a model. Hence, there must be a model for the theory T+not(G), but this cannot be the standard model (in which we can argue metamathematically that G is "true"). Thus, there is an interpretation of the theory (a model) in which G is false. This is not just "relevant", it is key to the Incompleteness Theorem and to the formal undecidability nature of the Goedel sentence. Magidin (talk) 18:21, 12 July 2016 (UTC)

Hmm, no, I wouldn't say "key". The incompleteness theorems are primarily proof-theoretic, not model-theoretic. By the completeness theorem, they have a model-theoretic counterpart, but while this is interesting in itself, it is not the main point. The impact of the theorems, in terms of making Hilbert's program appear hopeless and refuting at least the simplest versions of formalism and logicism, goes through the proof-theoretic formulations. --Trovatore (talk) 18:39, 12 July 2016 (UTC)

"However, the Gödel sentence of a consistent theory may be false in some nonstandard models of arithmetic." perhaps this is true but it would need context to make sense to anyone reading this article..it's just thrown in in the middle of discussion of a standard presentation of the theorem where G is simply true but undecidable by the system...the quick unexplained tangent is just like "what??"68.48.241.158 (talk) 18:51, 12 July 2016 (UTC)

I kind of agree with you, actually. The point should be treated somewhere, but I'm not sure it needs to be right there. --Trovatore (talk) 19:09, 12 July 2016 (UTC)

"Perhaps" it is true? Again, let me ask if you have read the SEP article, which has two paragraphs and a section title about the topic of nonstandard models! These are a very basic topic in logic, something that is seen in a first course, and we have an article Non-standard model of arithmetic. It is always the case that, if a consistent theory is unable to disprove some statement, then there is a model of the theory in which the statement is true. In the context of incompleteness these are important as examples of non-ω-consistent theories, which show that Rosser's trick is indeed important for the full proof of the theorem.

Frankly, I am finding this discussion somewhat surreal. To edit an article requires getting familiar with the basic topic of the article! I would recommend reading some recent sources - Peter Smith's book is excellent, Tokel Franzen's book is also quite good. The SEP article is brief, but it does show one model of an encyclopedia article about the subject. — Carl (CBM · talk) 21:46, 12 July 2016 (UTC)

Let's not fixate on the "perhaps". Yes, it's definitely true. But I kind of think 68 is correct that it's a bit jarring to suddenly drag in model theory at that point. Up to that point, no model theory has been mentioned. --Trovatore (talk) 21:53, 12 July 2016 (UTC)

yes, I'm not really suggesting it's wrong...just that it's sitting there in a vacuum or something...it's a writing style problem..the following section of this section, for lack of a better word, just sucks currently:

"...However, the Gödel sentence of a consistent theory may be false in some nonstandard models of arithmetic.

The truth of the sentence GF may only be arrived at via a meta-analysis from outside the system. In general, primitive recursive arithmetic proves the implication Con(F)→GF, where Con(F) is a canonical sentence asserting the consistency of F (Smoryński 1977 p. 840, Kikuchi and Tanaka 1994 p. 403)."

The first sentence of that next "paragraph" needs to be incorporated into the preceding paragraph, I firmly believe...and something just needs to be done differently with the technical references here.....68.48.241.158 (talk) 22:17, 12 July 2016 (UTC)

And I had solved the style problem at least to some degree by simply removing the other two sentences in a previous edit as I think the article can do away with that particular information (at least at this point in the article, etc)..68.48.241.158 (talk) 22:40, 12 July 2016 (UTC)

Prominence of mention of Cantor's diagonalization argument

I undid your [Leegrc's] most recent revision as I think it's simply too strong a statement to say 'based on' right in the intro...the article could use some more info about the 'diagonal lemma' and the similarities to Cantor etc but just not right at the top like that...go to that talk page if would like..68.48.241.158 (talk) 15:04, 11 July 2016 (UTC)

The lede currently indicates "Gödel's incompleteness theorems were the first of several closely related theorems on the limitations of formal systems" which is true but unprovable (hee hee). While I think it is important to credit Gödel for this giant work, it is nonetheless the case that he achieved it by standing on the shoulders of (other) giants. Gödel's insight was that Cantor's diagonalization argument could be applied to formal logic systems, if the logical statements about integers were themselves encoded as integers. I would like to see the briefest mention of Cantor in the lede; perhaps prefixing the above sentence with "Based upon Cantor's diagonalization argument," 𝕃eegrc (talk) 15:15, 11 July 2016 (UTC)

I just don't know it's fair/accurate to so strongly say 'based on' or some such right in the intro...it's possible a less strongly worded reference to Cantor could be worked in there...I do think the article itself could use a little more info about the influence/inspiration of Cantor and the "diagonal lemma" etc...there are a few others who watch this page closely and will be along to comment...68.48.241.158 (talk) 15:21, 11 July 2016 (UTC)

Perusing an online thesaurus, I see some alternatives to "based upon" such as "built on/from", "depending on", "hinging on", "standing on", "proceeding from", "grounded on/from", "rooted in", "anchored in". Do any of them put wind in your sails and/or are any of them at least in the right direction for what you are looking for? 𝕃eegrc (talk) 17:00, 11 July 2016 (UTC)

Idk it suggests something to me that isn't quite accurate, having it right in the intro like that...there's a section "relationship to liar paradox" or whatever....perhaps there could be a short section called "relationship to (or influence of) Cantor's Diagonal Lemma" or something...or maybe this info needs to be better developed/explained in the technical sections of the article....Gödel's work was influenced by/inspired by many things (including Cantor) but not sure the Cantor influence is so much more significant than all the others to necessitate such a nod right in the intro68.48.241.158 (talk) 17:17, 11 July 2016 (UTC)

but you're absolutely correct that he borrowed/used techniques developed by Cantor in the technical demonstration of the proof...I can't recall if he ever actually mentions Cantor in the paper itself..not that it matters..68.48.241.158 (talk) 17:20, 11 July 2016 (UTC)

How about "Leveraging Cantor's diagonalization argument, …"? Perhaps it is my background and my familiarity with Cantor's diagonalization argument and many of its diverse uses but, to me, these theorems of Gödel scream "Cantor" loudly enough to go into the lede. 𝕃eegrc (talk) 17:39, 11 July 2016 (UTC)

For now, I simply have to vote against any kind of statement like this...but I certainly encourage you to develop the info in the article itself (this will take more time/effort)...there are 3 or 4 other editors who watch this page closely who are fairly knowledgeable...they will be along...I strongly suspect they'll agree with me here (and they certainly don't always agree with me!) so this is why I encourage you to perhaps try the route of developing the article itself instead of the intro..68.48.241.158 (talk) 17:52, 11 July 2016 (UTC)

There is no need to acknowledge Cantor every time a diagonalization argument, a diagonalization argument-like argument or similar is used. It is part of the standard mathematical toolkit, and was so in Gödel's time too. Sometimes it is traditional to mention what is going on, naming a theorem, e.g. use of Zorn's lemma/use of the axiom of choice or the well-ordering theorem rarely goes without mention. But this is not the case here. "Using a diagonal argument" would perhaps be acceptable in some appropriate spot, not necessarily in the lead and without mention of Cantor. YohanN7 (talk) 09:59, 12 July 2016 (UTC)

Section: First incompleteness theorem

The text of the section "First incompleteness theorem" was getting out of hand. I have added a section header for some of the new material on truth, and arranged the section generally as follows:

• Statement of the theorem
• First mention of "Godel sentence" , and two existing paragraphs that mention what happens when you add the Gödel sentence as an axiom, and on the general idea of the proof. The latter paragraph needs help.
• Form of the incompleteness sentence - see below
• Truth of the Gödel sentence - note: refers to the idea of the proof mentioned already
• Existing section: "Meaning of the first incompleteness theorem" - note: refers to the truth of the sentence
• Existing section: "Relation to the liar paradox"
• Existing section: "Extensions of Gödel's original result" - on Rosser's trick

It is hard to rearrange the first three bullets into a different order, because each depends on things mentioned in the previous one. — Carl (CBM · talk) 23:04, 12 July 2016 (UTC)

still needs some shaping but, yes, this was a very good idea...68.48.241.158 (talk) 23:16, 12 July 2016 (UTC)

Following a suggestion of Trovatore I added a paragraph on the form of the Gödel sentence. There are many references on this point. — Carl (CBM · talk) 23:29, 12 July 2016 (UTC)

The section on the "Meaning of the first incompleteness theorem" needs work. It is a mixture of several ideas, and not in a sensible order either. — Carl (CBM · talk) 23:55, 12 July 2016 (UTC)

definite improvements overall..and I agree that next section is/has been problematic as well..68.48.241.158 (talk) 00:11, 13 July 2016 (UTC)

The smaller, more direct to the point sections at the beginning of the article are a big improvement...with such a big rewrite/restructuring there are at least many small style tweaks that will have to be worked out slowly....some later sections that seem a little not to the point and perhaps a bit repetitive of information include "EXAMPLES OF UNDECIDABLE STATEMENTS"/"FOUR VARITIES OF THEORIES"...68.48.241.158 (talk) 13:41, 13 July 2016 (UTC)

References

Would anyone mind terribly if I changed the referencing style to that in e.g. Cayley–Hamilton theorem? (Signature added after original post.) YohanN7 (talk) 10:02, 14 July 2016 (UTC)

(responding to proposal above..posted by YohanN7, I think): it certainly looks better and reads easier that way..so I would certainly support unless there's a particular reason why not..68.48.241.158 (talk) 13:34, 13 July 2016 (UTC)

It is mostly a policy, WP:RETAIN, that basically says don't fix it if it ain't broken (The link talk only about US vs British English, but the same principle is rather global.) YohanN7 (talk) 13:39, 13 July 2016 (UTC)

I'm confused...you want to change it, right?

Of course. I'll begin with what can be done without protests, templetizing the reference list. This opens up for having clickable citations, whether they are explicit (like (Franzén 2004, p. 112)) or ordinary footnotes.

One huge advantage of having references in this way is that it is so much easier to add citations. YohanN7 (talk) 13:48, 13 July 2016 (UTC)

yes, sounds good to me (I think you need to resign your first comment in this thread as it went away...so it kind of looked like I posted it)..68.48.241.158 (talk) 13:53, 13 July 2016 (UTC)

I'm sorry, I thought you were referring to the article's in-text citations...so nevermind..but I'm sure what you did is good and fine...68.48.241.158 (talk) 14:44, 13 July 2016 (UTC)

Yes, I would mind. This article uses regular Harvard-style referencing, which is a common style in actual published writing. It makes the citation for each sentence clear without having to look elsewhere (i.e. at a glance the reader knows which source is used). I would not support changing to the style of Cayley–Hamilton theorem. Also, I have found that, in general, the "clickable" references tend to be very fragile, and don't work as advertised except in very simple cases. — Carl (CBM · talk) 16:20, 13 July 2016 (UTC)

I'm all confused by this...I think he did something with the references at the bottom, after the article text..??68.48.241.158 (talk) 16:24, 13 July 2016 (UTC)

I think he converted some references to use citations. I don't really like that idea, either - the article mostly does not use templates, so probably the one or two that were added with templates should be changed to match the majority, rather than vice-versa. The citation templates also turn out to be inflexible except in simple cases; if there are reprints, multiple editions, etc., the citation templates can become an obstacle rather than a tool. — Carl (CBM · talk) 16:28, 13 July 2016 (UTC)

The templates are flexible enough.

1. A formal system is said to be effectively axiomatized (also called effectively generated) if its set of theorems is a recursively enumerable set (Franzén 2004).
2. A formal system is said to be effectively axiomatized (also called effectively generated) if its set of theorems is a recursively enumerable set .

• Franzén, T. (2004). An Incomplete Guide to its Use and Abuse. A.K. Peters. ISBN 1-56881-238-8. MR 2007d:03001. {{cite book}}: Check |mr= value (help); Invalid |ref=harv (help)

Editions, reprints and the like can be handled. One will get into trouble if one wants to list the translators brother in law and his wife, but there is nothing to prevent you from writing arbitrary text after the template or after the citation template in the footnote. Then, if one still has trouble, one should probably reconsider what belongs in the reference list and what does not.

Templates standardize the way references appear. Without them, a complete mess will result, especially with many references since most people don't know how to format a reference.

Moreover, they facilitate adding new inline citations that don't clutter the source text. They are clickable. A click takes you first to the right entry in the reference list and then (if the reference is complete in this respect), you can click your way to an online copy or a place to buy the item in question. As seen above, footnotes is not the only option.

Naturally, it should be used uniformly. I begun yesterday with the reference list (didn't touch the main body), and intended to do all of them, but that was reversed by CBM along standard info such as isbn numbers and editors. I truly don't understand why it is better to have the current "system". It sucks. YohanN7 (talk) 09:59, 14 July 2016 (UTC)

I do know the arguments for templates; I just don't find them as compelling as I once did. There's no general policy in favor of templates, and I think the current system is serving the article perfectly well. It has the benefit of being easy to type, and easy for other editors to pick up. In the end it's just a matter of preference, though, and so when there is a disagreement the usual rule of thumb is to keep the original version, which in this article was to have citations without templates.

At the same time I have no problem with adding ISBN numbers, and I'll go back and re-add any that I inadvertently removed. — Carl (CBM · talk) 12:12, 14 July 2016 (UTC)

Fair enough. I should have waited longer for replies after posting this thread. YohanN7 (talk) 12:23, 14 July 2016 (UTC)

arithmetic

I think the lede should specifically reference that we're talking about "arithmetical" relations here in a couple of those sentences...for a long while it had that parenthetical "(arithmetic)"...I suppose it's true that any relationship between them is inherently arithmetical...but for the sake of the general reader..

Also, this notion should probably be explained briefly somewhere as I don't think it's touched upon at all...these are Gödel's words from the Metzger translation:

"..there are in fact relatively simple problems in the theory of ordinary whole numbers which cannot be decided from the axioms." "theory of ordinary whole numbers" is synonymous with "arithmetic" here...his footnote for this sentence states, "ie, more precisely, there are undecidable propositions in which, besides the logical constants, there are no other concepts beyond addition and multiplication, both referred to natural numbers and where prefixes can also only refer to natural numbers." later in the paper he briefly states, "a relation is called arithmetical if it can be defined solely by the means of addition and multiplication applied to natural numbers.." (keep in mind that the inverses (division and subtraction) can be stated via the concepts of addition and multiplication in these systems..) 68.48.241.158 (talk) 16:35, 13 July 2016 (UTC)

"all truths about the natural numbers" seems vague...there are probably truths that aren't arithmetical...idk, a really dumb example is that the symbol for the number 5 can be found on my mailbox...idk, point is the general reader might not no what this means..68.48.241.158 (talk) 16:47, 13 July 2016 (UTC)

(^^I'm operating under the assumption that I'm basically talking to two other editors who understand the article edit activity that led to these threads...so these threads are likely incomprehensible to the uninitiated...I can clarify the issue better for others, if requested)..68.48.241.158 (talk) 19:04, 13 July 2016 (UTC)

Role of Self-Reference and Franzen

This section needs to go imo...it adds little or nothing and the individual it's quoting is simply not particularly notable (ie his opinion, whether right or wrong isn't notable enough to include...unlike Wittgenstein whose opinions in this area (though mostly wrong) are most certainly notable...he's an obscure academic who wrote yet another fairly obscure book that attempted to explain the theorems to the general reader...idk how much he should even be cited in the article itself...I think the article can simply cite Gödel's paper itself for most of it's content/assertions...or is this against the rules?...like the above quotes from his paper about arithmetic...could we just cite him or do we have to cite someone else explaining it?? (I don't mind him being referenced briefly in the section just above along with Rebecca Goldstein etc, however)...68.48.241.158 (talk) 17:38, 13 July 2016 (UTC)

^as stands, I feel it's basically exactly analogous to Hewitt's desire for commentary by whoever it is he wants commentary by.....68.48.241.158 (talk) 17:53, 13 July 2016 (UTC)

Generally secondary sources are better for our purposes than primary. Certainly Gödel's own views are relevant, especially for the historical angle, but ideally what we're looking for is the conclusions that experts have come to in the intervening decades, with lots of time to think about it and polish their formulations.

Franzén is a useful source because he was a very smart guy who spent a lot of time analyzing the import of the theorems and making them accessible, but unlike the popularizers, he was personally an expert in the field.

I don't entirely like his take on this aspect, as I don't think the theorems are self-referential in the first place, so there is no need to strain to find formulations that are not. But I do think he's a good source. --Trovatore (talk) 18:50, 13 July 2016 (UTC)

depends what you mean, I guess..the notion of self-reference plays in with the meta-analysis from outside the system that discovers the truth of the proposition...in any event, this stand-alone section seems entirely dedicated to a fairly unimportant, idiosyncratic opinion of a fairly non-notable person...68.48.241.158 (talk) 19:10, 13 July 2016 (UTC)

The more I look at that section, the odder and more inappropriate it looks to me...it's an undeveloped stand alone section dedicated to an idiosyncratic opinion statement by a contemporary, non-notable academic...I'm going to delete it...per my reasoning and apparent at least soft support of Trovatore....if CBM or someone else wants to reinsert it please come here and explain....68.48.241.158 (talk) 00:10, 14 July 2016 (UTC)

I "approved" the edit, but that term was a bad choice by the software engineers. In general, I will approve edits even when if I have no opinion or possibly even disagree with them, so just because I "approved" them should not be taken as a comment other than that I don't want to see them "pending". (This is why sometimes I may "approve" a revision and then undo parts of it.) — Carl (CBM · talk) 00:38, 14 July 2016 (UTC)

fair enough, thanks..68.48.241.158 (talk) 00:48, 14 July 2016 (UTC)

I'm not sure I agree with the cavalier dismissal of Franzen by 68/anonymous editor. Franzen was a well-regarded expert in the field who wrote several books on the subject, both in terms of addressing a general audience and in terms of a mathematical one. I would put him on par with Raymond Smullyan in terms of reliability. The entire tone above sounds to me like "I don't like this quote. Since I've never heard of him, he is not a notable source and this should be deleted." The comments on self-reference are at least somewhat relevant (probably more), especially when you have folks in this talk page who imply and argue that the statements are self-referential and hence should be "disallowed". Magidin (talk) 03:29, 14 July 2016 (UTC)

this single opinion/commentary quotation simply doesn't warrant its own stand alone section imo...it's also basically repeating information from an earlier section in the article (as the explanation sentence below the quotation even says...imo work it in up there very briefly, if deemed worthwhile..but not with the whole long quote...nobody's quoted at such long length in the article...not Bertrand Russell, not Wittgenstein, not Godel himself...but "Torkel Franzen"??? It's the same thing as Hewitt wanting some long, also odd quotation by whatever unimportant person he wanted commentary by....68.48.241.158 (talk) 12:19, 14 July 2016 (UTC)

Yes, I think Franzén is a fine source. Not as many people know him as know (say) Hofstadter, and he's not as good a writer (that's a high bar), but he has a much deeper and more solid understanding of the subject matter.

In context he doesn't seem to be saying that the sentences in question are self-referential, just that people perceive them to be and that misunderstandings result from this. I agree that's probably accurate; I just have a different strategy, and would prefer to tackle that misunderstanding at its source, by showing that the sentences are very ordinary (if long-winded) assertions about natural numbers, not about the sentences themselves.

But I obviously can't demand that Franzén follow my strategy. I think the material is reasonable to include. I'm not sure it merits its own section, though. --Trovatore (talk) 04:59, 14 July 2016 (UTC)

I agree it at least doesn't seem to merit its own stand alone section, at least as currently constructed..68.48.241.158 (talk) 11:36, 14 July 2016 (UTC)

Of course there is beaucoup literature that describes the Goedel sentence as self-referential, so it won't work for us to simply claim it isn't. Of course, the self-reference is indirect, just as with the recursion theorem in computability theory. And the Goedel sentence is just a $\Pi _{1}^{0}$ sentence of arithmetic, like the statement of Goldbach's conjecture and like many other statements about elementary number theory. Franzen's text is quite high quality and "mainstream" within the limited number of scholarly secondary sources directly about the incompleteness theorems. In the end, the Goedel sentence relies more on a kind of diagonalization than it does on self-reference, but finding a source that clearly expresses that may be challenging.

I don't mind the current set up, in which the article points out that the Goedel sentence indirectly refers to itself. On the other hand, it would be worth pointing out that the underlying phenomenon does not really depend on self-reference, because when we view the Goedel sentence as just a formula of arithmetic, forgetting its intended meaning, there is no self-reference there. Goedel mentioned this fact in print, as well, in responding to Wittgenstein, as we quote. For that subject, I am sure we can assemble something from a few sources - I suspect Peter Smith's book also comments on the issue. — Carl (CBM · talk) 12:50, 14 July 2016 (UTC)

^and some of this speaks to confusions even otherwise learned people like Hewitt have about this topic imo..indeed, perhaps some of the new phrasing in the article along these lines will be helpful to readers..68.48.241.158 (talk) 13:11, 14 July 2016 (UTC)