Talk:Consistency

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: Start Class Mid Priority  Field: Foundations, logic, and set theory
Please update this rating as the article progresses, or if the rating is inaccurate. Please also add comments to suggest improvements to the article.

Contents

[edit] Gödel's incompleteness theorems

The info on the page about Gödel's incompleteness theorems is good, but I think it would be good to define consistency in its own right, and explicate it in more detail here. What I put here right now is pretty weak, though. —Preceding unsigned comment added by 140.142.182.182 (talk)

[edit] Whole lot of problems

The list starting "systems proved to be consistent" is simply bad and evidence of confusion. If it was changed to: systems that are complete wrt. a model (in the sense of maximal complete set) this would work, but then wouldn't fit in the category. The leading paragraph isn't great either. I've shifted the page from Consistency to Consistency proof, changed Consistency into a disambig page, and rewritten it. The previous text was:

In mathematics, a formal system is said to be consistent if none of its proven theorems can also be disproven within that system. Or, alternatively, if the formal system does not assign both true and false as the semantics of one given statement. These are definitions in negative terms - they speak about the absence of inconsistency. Formal systems that do admit contradictions suffer a semantic collapse, in the sense that deductions in them cannot truly be assigned any significant content, by schemes that apply across the whole system.

To add:

  • Systems proved to be consistent
  • Systems not proved consistent
    • First order Peano arithmetic (from a system no stronger than Peano arithmetic)
  • Systems that cannot be proved consistent

I'm adding:

  • Intro to incompleteness theorems
  • Disccussion of relative consistency proofs
  • Complete systems
  • Self-verifying systems
  • Essentially incomplete theories

I plan to add later:

  • Pi-0-1 nature of consistency statements and relevance to Hilbert's program (see Proof theory for an outline)
  • Shared potential of essentially incomplete theories
  • Discussion of provability logic

[edit] Alternate senses of Consistancy

The only sense of "Consistancy" given in the article is the lack of explicit contradiction. This applies only to Classical Logic systems. (They this doesn't work, for example, for Graham Priest's Paraconsistant logics.)

This sense also fails to work at all in systems without an explict negation. (Since by that definition any such system can't generate a contradiction, since they have no negation).

However, senses that are functionally equivalent to the standard one in systems with negation, and still work in positive propositional logics, have been around for more than 70 years... Hilbert (about the time he was playing with positive propositional logics) gave a number of definitions that worked for his use.

His "Absolute Consistancy" (for example) just says that if a system can prove anything, it is inconsistant. If there are Well Formed Formula in a system that the system can not prove then it is consistant. (For traditional systems, if the system contains a contradiction then it can prove anything. And if there are WFF's in the system that it can't prove then it must not have a contradiction, since a [traditional] system with a contradiction can prove anything.)

As an example, Feys, in his 1965 book "Modal Logic" (published posthumously) established the consistancy of a number of Modal Logic systems by showing that there were WFF's in those systems that those systems could not prove. [Really short slick proofs, by the way].

I would personally find it a better if a more general sense of consistancy were used that applied to more than just classical logics. (Note that Wikipedia has a Paraconsistent logics page, so this page doesn't cover a notion that even applies to all the logics on the Wikipedia pages, much less the logics in the literature.)

Nahaj 02:55, 31 October 2006 (UTC)

Yes, this applies only to classical logic. But most mathematicians and scientists assume classical logic unless one explicitly specifies another kind. Virtually all serious work on mathematics is done in classical logic. Classical logic is what is needed to deal with the real world. Other forms of logic are just games or fantasies. JRSpriggs 10:38, 31 October 2006 (UTC)
And when one specifies another kind, this definition doeesn't work. All others being "Games or fantasies" is not a nice thing to say. I have to wonder if you are familiar with the others. But fortunately, there are others on wikipedia that don't agree with you that have put up pages on these other logics. Too bad this page will have (without a statement to that effect) narrow information that doesn't apply to them. Nahaj 01:44, 1 November 2006 (UTC)

[edit] What is this article about?

I proposed here that Hilbert's second problem should be covered in its own article, separate from this one. Please discuss it there. CMummert 14:36, 5 January 2007 (UTC)

[edit] WikiProject class rating

This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 03:52, 10 November 2007 (UTC)

[edit] Bunch of links

Just so that everyone knows, Gregbard recently linked hundreds of occurrences of the word "constistent" to this page. Many of the links had nothing to do with mathematical logic. I am in the process of removing those links which do not fit this context. Another avenue might be to expand the scope of this article to give a more notional definition of consistency, applicable outside the domain of mathematical logic. The current page can then be moved over to consistency (logic) or the like. At any rate, it seems fairly clear to me that, unless there is a formal system close at hand, then linking to an article on consistency in logic flies in the face of WP:CONTEXT. silly rabbit (talk) 14:27, 1 May 2008 (UTC)

Where does your idea come from that the world revolves around mathematical logic, and every other link needs to go? Consistency means something does it not? Is there some other concept that is being meant? NO. It's the same concept rigorously defined. It happens to be a very important concept in philosophy. Furthermore, this article has almost none of the philosophical account of consistency even though it belongs in this article, not separated out. This is why it was moved to "consistency" rather than the narrower "consistency proof". I have done the same thing for many other articles in logic, but I have never had this problem. You seem to believe that only mathematicians need concern themselves here, and anything else should be split out. This is called disintegration, and it causes there to be two crappy articles, rather than one comprehensive article. The motivation seems to be that the mathematicians simply can't work well with others interdisciplinarily. Please relent from the narrow view. Be well, Pontiff Greg Bard (talk) 19:41, 1 May 2008 (UTC)
If it is a very important concept in philosophy, why does the Stanford Encyclopedia of Philosophy not have an entry for it?  --Lambiam 22:50, 2 May 2008 (UTC)
You'll have to ask them. The Cambridge Dictionary of Philosophy has a full page entry. Hmm, Stanford v. Cambridge. The Stanford website also has plenty of material on consistency in other articles. Pontiff Greg Bard (talk) 23:03, 2 May 2008 (UTC)
The CDP entry starts with a single brief sentence on the notion of consistency in philosophical logic, and then spends the rest of the article on the (different) notion of consistency in mathematical logic.  --Lambiam 15:40, 3 May 2008 (UTC)
To respond to Gregbard's original reply, it is fine if you link occurrences of the bare word consistency, as long as it really will benefit the target reader to have the term linked to a technical article on mathematical logic. Many of the links were not used in this sense, and linking them here creates the misleading impression that the word is being employed in a technical, rather than ordinary language, manner. This has absolutely nothing to do with disintegration. I have left all links alone where the context was appropriate. Perhaps we should bring this issue to the attention of Wikipedia:Wikiproject Integration so that members of that project can weight in on this dispute. silly rabbit (talk) 01:35, 3 May 2008 (UTC)
I am fine with many of those reverts. We have covered many (but not all) of the strictly technical references to that term. No, this link issue is not about disintegration. However, the article may evolve to cover more ordinary aspects of it. For one thing I presume that people mean what they say on the WP, so it is more reasonable to interpret many of those "ordinary" uses as identical to this topic at least. Then whether or not it is believed that it rises to the level to where it should have a wikilink is a matter of priority.
Shocker:I have been making links to all kinds of terms for quite a while now, and I have never had such a strong response. I think any term that is included in the template:logic at least should be incorporated as wikilinks wherever it is appropriate. There are reasonable cases such as "ordinary" uses of the word theorem; and more unreasonable cases such as relation (mathematics) where is would be unreasonable to wikilink every relationship in the wikipedia. I have been using my judgment on those, and I think I have been doing just fine and you are over-reacting about it. We disagree about the priorities, but not the principles in wp:overlink. Pontiff Greg Bard (talk) 03:59, 3 May 2008 (UTC)
You may notice that I have left intact all links on notional references to "logically consistent", despite the inappropriateness (in my opinion) of many of these. I agree with your sentiment that an author should be prepared to mean what she says. silly rabbit (talk) 18:38, 3 May 2008 (UTC)