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- 1 Isomorphic
- 2 Consistancy
- 3 Perhaps make it more accessible...
- 4 "Model" as a term of art in logic
- 5 Merge "Axiomatization" into "Axiomatic system"
- 6 Merger of "Axiomatic system" and "Formal system"
- 7 WikiProject class rating
- 8 Either-or?
- 9 Outside of mathematics
- 10 Completeness
- 11 Incomplete definition
"An axiomatic system for which every model is isomorphic to another is called categorial" — Is this correct? I would expect "...every model is isomorphic to every other". In the original, there can be multiple, distinct isomorphy classes, while in the latter by transitivity there is only one, so I see a connection to completeness. It is hard to imagine how the property of a model being "isomorphic to another" can be meaningful in any sense, not least since isomorphy is reflexive...
The article says:
An axiomatic system is said to be consistent if it lacks contradiction, i.e. the ability to derive both a statement and its negation from the system's axioms.
Sigh Yet another wikipedian place where "consistant" is assumed equal to "no contradiction". Axioms systems that have no negation can never generate a contradiction. Yet such a system can be either consistant or inconsistant.
In traditional PC contradiction is bad because it allows you then prove any statement whatsoever. It is *that* property that makes contradiction fatal to the system. But it is not the only property with that sort of fatality. The system consisting of only the axiom "p" is inconsistant because you can generate (by substituting for p) any statement whatsoever.
Maybe something closer to:
An axiomatic system is said to be consistant if there are things it can prove, and things that it can not prove. Contradiction (proving something and its negation) is an example of a property that makes a system inconsistant.
This, at least, is true for systems without explicit negation.
- I think you are trying to describe the property of explosion, i.e. of being able to prove every proposition. In non-paraconsistent logics, contradiction leads to explosion. However the earlier editor is still correct to say that contradiction means able to prove p and not p for some p.DesolateReality 14:07, 21 July 2007 (UTC)
- But still wrong to say that consistent is the same as lack of contradiction. Even as early as the late 1940's there were logics (S0.5) that had explicit contradictions but didn't exhibit "explosion" [to use the terminology of the parconsistent crowd] and are generally called consistent. By the way, if a system can't prove "p" it is traditionally called "Hilbert consistent". Hackstaff's "Systems of Formal Logic" has a good discussion of the various notions of consistency running about.126.96.36.199 (talk) 20:38, 16 February 2011 (UTC)
- I don't see those systems as "consistent". The usual definition, in any case, is for first-order theories, not for paraconsistent (but "inconsistent") theories. In Hackenstaff's terminology, we are talking about "Aristotle consistency", not "absolute consistency". But in the usual setting all four notions discussed by Hackenstaff are equivalent, of course. — Carl (CBM · talk) 02:29, 17 February 2011 (UTC)
Perhaps make it more accessible...
This page is written in very complex terms for people who do not understand high levels of mathematics. Can we edit this page (and until they are complete, tag it) so the language it is written in is more accessible to a wider group of people?
"Model" as a term of art in logic
I changed the wording of the first sentence under "Models" to say "model" instead of "mathematical model" because the former is a term of art as well as standard terminology in mathematical logic. --188.8.131.52 16:22, 5 August 2006 (UTC)
Merge "Axiomatization" into "Axiomatic system"
I'm suggesting placing the content of Axiomatization under the section of Axiomatic method in the present article Axiomatic system. This will add bulk. The two articles do not seem to be very distinct from each other.--DesolateReality 03:53, 22 July 2007 (UTC)
Merger of "Axiomatic system" and "Formal system"
I suggest a merger of these two pages. I recognize a formal system as a special case of an axiomatic system. There can be axiomatic systems in political philosophy or ethics (I got this from the commentary Axiomatic system#Axiomatic method) which are not in a strict formal alphabet nor use a formal logic as means to get theorems. The section Axiomatic system#Properties such very much an appropriate description of mathematical formal systems and so the merger will make this link clearer. The commentaries Axiomatic system#Axiomatic method and Formal system#Formal proofs can also be tightened up and expanded with such a merger.
I have removed the merger tag. read comments at Talk:Formal system#Merger of "Axiomatic system" and "Formal system" --DesolateReality 05:36, 8 August 2007 (UTC)
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 03:48, 10 November 2007 (UTC)
"An axiomatic system will be called complete if for every statement, either itself or its negation is derivable."
Wouldn't that be the definition of a "complete and consistent" axiomatic system? That's restricting the definition of "complete". Can't it be complete and inconsistent? Either-or implies non-contradiction... —Preceding unsigned comment added by 184.108.40.206 (talk) 04:40, 2 February 2009 (UTC)
Outside of mathematics
I think the statement "as shown by the combined works of Kurt Gödel and Paul Cohen, impossible for axiomatic systems involving infinite sets" is somewhat misleading. Many axiom systems are complete and could be thought of as involving infinite sets (e.g. algebraically closed fields of characteristic 0). I assume the result being referred to is the independence of CH from ZFC. Maybe better to give this an explicit example. Impossibility would seem to point more toward Godel's incompleteness theorems, which I don't think Cohen's work relates to.
It might be good to qualify "impossible" as well. Complete sets of axioms exist for any model - just take the set of all statements true of that model. The incompleteness theorems talk about the impossibility of computable sets of axioms.
- Unfortunately, your comment was not noticed when you made it. You're completely correct. — Carl (CBM · talk) 12:46, 21 June 2010 (UTC)
The article addresses axioms without saying what they are about. The article should link to primitive notions. For example, William Alfred Thompson wrote in The Nature of Statistical Evidence (Springer Lecture Notes in Statistics #189), page 10:
- The axiomatic method introduces primitive terms (such as point and line) and propositions concerning these terms, called axioms. The primitive terms and axioms taken together are called the axiom system Σ.