Talk:Consistency proof
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[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. —The preceding unsigned comment was added by 140.142.182.182 (talk • contribs).
[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)
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- 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)
