Talk:ω-consistent theory

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[edit] lowercase

Samuel Wantman left an edit summary asking why "Omega" shouldn't be capitalized. I'm not claiming it's completely clear-cut, but here's my argument: Ideally the article should be at ω-consistent theory, with the lower-case ω. If we use the Greek letter, then the capitalization is very important; the phrase "Ω-consistent theory", though (as far as I know) having no standard meaning, could imaginably be taken to mean "consistent in Woodin's Ω-logic", an immensely stronger notion.

The part that's a little less than clear-cut is the association "Omega/Ω :: omega/ω". This is standard, though, to anyone who uses TeX. --Trovatore 17:05, 31 August 2006 (UTC)

I've changed the note to reflect that ω-consistent theory is the ideal name. -- Samuel Wantman 18:49, 31 August 2006 (UTC)
No, it isn't. Gene Nygaard 02:05, 3 December 2006 (UTC)

[edit] move request

ω-consistent theory is the standard name used in the literature. The reason I didn't use it when I started this article is that it would have rendered "Ω-consistent theory", which is quite wrong. However {{lowercase}} now provides a javascript hack to render the title correctly in javascript-enabled browsers. --Trovatore 06:59, 25 November 2006 (UTC)

This does not look controversial to me, so I moved the article. -- Jitse Niesen (talk) 08:19, 25 November 2006 (UTC)
Can a similar hack work for the talk page? It's kind of weird that they don't match now. -GTBacchus(talk) 08:26, 25 November 2006 (UTC)
Answered my own question: it seems to work. -GTBacchus(talk) 08:27, 25 November 2006 (UTC)
Still rendered as "Ω-consistent theory" when I click 'history' or when I'm editing this page now. Confusing like woah. Haukur 09:05, 25 November 2006 (UTC)
Yes, that is unfortunate. I'm trying to gather support for a developer change that would turn off the automatic uppercasing when the first letter of an article name is non-Latin (or, at the very least, if it's Greek). Haven't seen much movement yet. Maybe you'd like to comment at Wikipedia:Village pump (technical)#Uppercasing of non-Latin letters. --Trovatore 05:26, 26 November 2006 (UTC)

Moving the page for the sole purpose of making the wrongly placed "lowercase" template justified is sheer nonsense. Gene Nygaard 02:06, 3 December 2006 (UTC)

And, lest anybody is dumb enough to be fooled by the Java-script shenanigans with the display on this articles page, just go follow the links to the one real non-stub category in which this article can be found, and come back and tell us exactly what sort of nonsense you see when you get there. Not only what you see for the article name, but also what letter you find it listed under.
Just where the fuck do you find it, anyway? Off in oblivion, somewhere after the Z.
I'll hold off on fixing the sort key properly until at least a few of you get a chance to see how you are squirreling away this information, hiding it so that is is unfindable. Gene Nygaard 02:13, 3 December 2006 (UTC)
Thanks for the reminder about search keys, Gene. Those are way too often overlooked, and it's good that you're conscientious about it. -GTBacchus(talk) 02:37, 3 December 2006 (UTC)
One of the biggest problems is that users of this template get hoodwinked into some erroneous belief that it is something other than a gimmick that just works on the display of this one page at the most, and often not even that. Gene Nygaard 02:51, 3 December 2006 (UTC)


The purpose of the move was not at all to justify the "lowercase" template. I don't like that template at all when it's unnecessary, and even when it is necessary, it's still a hack.

Rather, the purpose was to put the article at its correct name. Using "omega", spelled out in Latin letters, is just wrong. I challenge you to find any logic text with the phrase "omega-consistent" or "omega-consistency" in its index. You won't find it. --Trovatore 02:40, 3 December 2006 (UTC)

I have a better idea. I challenge you to show me any index from any book which indexes this concept after the Z's. Gene Nygaard 02:48, 3 December 2006 (UTC)
Ok, what we learn from this pair of challenges is that we don't have the ideal situation here, so something has to give. Is it worse to have the article titled and indexed as "omega-consistent theory", which is wrong, or to have the title show up correctly while it's indexed as "Ω-consistent theory", which is also wrong? I prefer having a correct title show up at the top of the page, by whatever kind of shenanigans. Either way, it should certainly be indexed under the letter 'O', but I don't believe anybody's arguing that it should come after 'Z'. That was the result of carelessness, and we're lucky to have Gene here to catch us on that. -GTBacchus(talk) 02:55, 3 December 2006 (UTC)

It would be trivial to index it under "O" by using the piping trick, if that were the correct thing to do. I'm not convinced that it is; I think non-Roman starting characters such as ω should be indexed separately. —David Eppstein 04:58, 3 December 2006 (UTC)

I agree. In many cases, if you wanted to interleave Greek letters with the Latin ones, it's not clear which one you would use, or the correct result is non-obvious. An article that starts with φ would go under F, I suppose, but where do you put one that starts with χ? Under H, seems to be the most defensible answer, but not too many people will think of looking there. And there's simply no letter that corresponds to Ψ or θ. --Trovatore 05:18, 3 December 2006 (UTC)
I don't know. My analysis books index "σ algebras" under 's', "ε-close mappings" under 'e', "ν*-measurable sets" under 'n', etc. I'd like to see a book that indexes concepts beginning with Greek letters separately from those starting with Roman letters. As for Ψ and θ, they're alphabetized as "Psi" and "Theta". -GTBacchus(talk) 09:37, 3 December 2006 (UTC)
Of course they are. Trovatore's suggestion of φ under f rather than under p or χ under H rather than under c is silly. And even if he/she doesn't "find any logic text with the phrase 'omega-consistent' or 'omega-consistency' in its index" but does somehow find "ω-consistent" or "ω-consistency" in the index, the latter will be found in the very same place in the index that the former would be found. Gene Nygaard 11:41, 3 December 2006 (UTC)
"He" of course -- otherwise I'd be Trovatrice, obviously. On looking through a couple of indices, it does seem that Gene is correct about how these things are usually indexed. I don't think it's the most logical way, as it seems to give undue weight to the Latin spellings of the letters, but it does have the advantage of being easy to remember, I guess. We should probably have a WP-wide discussion as to how these things should be indexed (or at least math- and physics-wide; it seems unlikely to come up in the rock album and pornstar categories). --Trovatore 18:17, 3 December 2006 (UTC)
Actually, it is more likely to come up in rock album categories, at least. But that is not only the way it is done (and for that reason alone not something for Wikipedia to do differently), but it is the logical way to do it. Actually, this is a lot like "pascals" and their symbol "Pa". Just as many publications will use both "Pa" and "pascals", there are many which would use both "ω-consistent" and "omega-consistent". It isn't an either/or proposition. Gene Nygaard 19:12, 3 December 2006 (UTC)
Have we seen an example where a publication uses "omega-consistent"? At any rate, I've got a copy of Cohomology of Groups by Kenneth S. Brown in front of me, and I noticed something odd about its index. The regular index puts both ∂-functor and δ-functor (in that order) under 'D', which seems normal to me. Then, the index of symbols, which is also alphabetical, alphabetizes 'Γ' and 'γ' under 'G', 'ε' under 'E', 'ρ' under 'R', and 'Σ' under 'S', but... for some reason 'χ' is alphabetized as an 'X'. Odd, eh? -GTBacchus(talk) 19:24, 3 December 2006 (UTC)
Leblanc, Roeper, Thau, Weaver, "Henkin's Completeness Proof: Forty Years Later", Notre Dame Journal of Formal Logic,, vol 32, no. 2, Spring 1991 [1] Gene Nygaard 19:46, 3 December 2006 (UTC)
OK, you got one. I think at any rate you'll find it's less common. Even Hofstadter uses "ω-consistent"; I checked. I thought he'd be among those more likely to say "omega-consistent", being a popularizer.
I disagree that alphabetizing by the Latin names for letters is the logical way to do it. The Latin equivalent of the letter φ is not the word "phi"; that's just the name for the letter. Rather, the Latin equivalent of the letter φ is the letter f. Most Greek letters have more-or-less equivalent Latin letters, but θ and ψ clearly don't, χ and ξ are problematic, ω and ο map to the same letter, and ι maps to two different letters (i and j). --Trovatore 20:05, 3 December 2006 (UTC)
But it isn't being used as a letter. It is being used as a symbol. The φ isn't pronounced "f"; it is spoken as "phi". The "omega" here is likewise spoken that way; it isn't merely a letter, a component used to construct words and being part of the pronunciation of those words, but rather a symbol. Gene Nygaard 20:26, 3 December 2006 (UTC)
Yes, of course it's being used as a symbol. But "omega" is not the symbol; it's the name for the symbol, the same way "zee" (or "zed" if you prefer) is the name for the letter z. You wouldn't write "the zee-axis", and similarly you shouldn't write "omega-consistent", not if you have a choice. --Trovatore 20:32, 3 December 2006 (UTC)
The difference with z is that it is itself an English letter. We do, however, write em dash and en dash, and delta-vee is quite common (13,100 google hits). That's just scratching the surface, of course. Gene Nygaard 03:51, 4 December 2006 (UTC)
And "wye delta" (43900 Google hits). There is, of course an inherent clash in mixing together symbols and spelled out words. It's like writing "coulombs/kg", something expressly against the rules of many style guides.[2] Gene Nygaard 04:07, 4 December 2006 (UTC)
The fact that z is itself an English letter makes no difference. We have Unicode now. Learn it, use it, love it. --Trovatore 06:39, 4 December 2006 (UTC)

Ok, so... I've got a reliable source that indexes Greek-letter words along with corresponding Roman letters, where the correspondence is in whatever logic puts 'chi' in the 'X' section. What other kind of examples have we got in sources regarding indexing of words beginning with Greek-letter symbols? Examples involving ω-consistentcy are preferable to the other kind. Smoryński's Self-reference and Modal Logic might be a good place to start, being the article's sole reference. -GTBacchus(talk) 07:22, 4 December 2006 (UTC)

[edit] Yes, I know It's Weaker

That's why it was phrased so carefully, as a rule of thumb, not as an equivalent. The example on this page is then not an example of omega inconsistency, just of unsoundness. You should put a real example of omega inconsistency. I was going to put that up today, but you reverted my last, carefully written comments.Likebox 16:52, 16 October 2007 (UTC)

A rule of thumb for what: for recognizing ω-consistent theories? Then a suitable rule of thumb is "all decent theories are ω-consistent". [Unless they are theories of nonstandard analysis. 12:45, 18 October 2007 (UTC)] Your "rule of thumb" was not only inadequate (as it mixed together quite different consistency principles), but also significantly more complicated than the simple and perfectly clear definition of ω-consistency itself, which defeats the whole purpose of a rule of thumb. That's why I reverted it.
There is nothing wrong with the example, it is a good example of ω-consistency. Σ1-ill theories are ω-inconsistent, it's the converse which breaks (badly). Nevertheless, I've added a Σ1-sound example as well for comparison. I've also included more info on other soundness-based principles, as well as ω-logic, to make it more clear where ω-consistency stands. Like it or not, ω-consistency does not have any computational explanation, so please do not try to make one up by confusing it with other properties.
By the way, was 71.176.115.195 you, logged out by accident, or is it a coincidence? EJ 13:21, 17 October 2007 (UTC)
I meant, yes, 71.176.115.195 IS me...
No-- just bad internet service! Sorry, I think it's clear when it is me. The only reason I put it up is because I wanted to link to Godel's incompleteness theorem, and there the example is sigma-one-unsound, and that's the essential point. Since Godel introduced the concept, he had sigma-one-soundness in mind, not necessarily oracle-soundness. What you did is fine by me though.Likebox 18:59, 17 October 2007 (UTC)
Yes. I think it's best to use Σ1-soundness directly when stating or explaining the first incompleteness theorem, and to avoid ω-consistency altogether (or to reduce it to a historical footnote). -- EJ 12:45, 18 October 2007 (UTC)
Wait a second, I just noticed you said that ω consistency does not have a computational explanation. That's not true. It's just that you need to add an oracle to your computer. That's the arithmetic-hierarchy/Turing hierarchy equivalence of Post. There's nothing in mathematics which does not have a computational explanation.Likebox 19:01, 17 October 2007 (UTC)
But you said that in the article too. Thanks. Wow. nice job!Likebox 19:07, 17 October 2007 (UTC)
Thanks.
I don't quite agree with the oracle business. If you reformulate "an arithmetical formula" as "a Turing machine (or whatever) with an oracle from the arithmetical hierarchy", it does not really explain anything. It merely shifts the complexity of the problem to the oracle, and wraps it in an additional layer of stuff. It may superficially look like computation, but the real thing is still happening in the oracle. It may be kind of useful to do this for restricted classes of formulas like Σn (because the halting problem provides an existential quantifier for free, which reduces the complexity by one level), but in the general situation it is IMHO pointless. -- EJ 12:45, 18 October 2007 (UTC)
Pointless, meaning mathematically equivalent, I agree. Pointless meaning philosophically equivalent, not exactly. They are somewhat different sources of intuition about what kind of arguments to make in the future. You don't have to think of the oracle as a computer. The oracle can be approximated by a large random-access CD-ROM with imperfect data which has been previously computed by someone else. So if you want to look at the properties of finite computations you will have different ideas when thinking about oracles vs. thinking about classes of formulas. But, philosophical quibbles aside, I think all the article is very good now. I would like to translate your sigma-one sound omega-inconsistent statement into an oracle computation, though, even though it's "pointless".Likebox 14:51, 18 October 2007 (UTC)

[edit] Notation

The notation \Sigma^m_n is used without definition. Can someone please provide one? -- Hairy Dude (talk) 18:02, 16 November 2007 (UTC)

There is a prominent link to the arithmetical hierarchy in the article. Just click it. -- EJ (talk) 10:05, 20 November 2007 (UTC)

[edit] Numerically insegregative

As currently phrased in the article, it would seem that the property of being "numerically insegregative" introduced by Quine is a synonym of "ω-inconsistent". But it is actually a stronger property, meant to address something Quine viewed as unsatisfactory with the concept of ω-inconsistency, which he considered misleading. The precise definition of Quine's revised concept is rather complicated, and implies that the logic underlying the theory cannot express, in some sense, the predicate "n is a natural number".

The result by Smoryński mentioned later on ceases to hold if "ω-consistent" is replaced by "numerically segregative". Thus the identification of the concepts is problematic.

Since Quine's revised concept appears not to have caught on – I can't find any references to it other than in Quine's publications or reviews of them – and is rather abstruse, I propose that the references to it be omitted. Alternatively, should someone (who really understands the issues involved!) feel inclined to produce the necessary prose, there could be a separate section stating something like that Quine considered the concept of ω-inconsistency misleading and came up with a stronger alternative.  --Lambiam 16:01, 20 May 2008 (UTC)

I would agree completely, were it not for your taking the opportunity to slight me. The term appears to have at least some life outside of reviews of Quine: link. It appears to be a consensus in the literature that this is a more appropriate term for this concept. If it actually describes a subtly different concept, then I would say that it doesn't need a separate article, but rather, as you say, there should be a section on it. Do you feel competent to that, or should I work on it?
Pontiff Greg Bard (talk) 20:45, 20 May 2008 (UTC)
Should I feel slighted by the ascription to me of taking opportunities to slight others? When I searched for "numerically segregative" on Google books, the Tymoczko book did not show up.[3]
I don't understand the reasoning behind Quine's complaint that the definition and concept of ω-inconsistency are misleading. In particular I don't get what he means by "failure of existence of needed classes". After all, whether the class comprised of precisely the natural numbers exists should be independent of any logical system. If it is the case that the logical characterization of N in a system "thus afflicted" can sometimes be repaired, as Quine suggests, how does this formal manipulation cause a class to pop into existence that failed to exist before? I think that if this is described in a separate section, it should be done by someone who does understand Quine's reasoning. I maintain, though, that the issue is abstruse and of minor importance.  --Lambiam 23:22, 21 May 2008 (UTC)