Talk:Axiomatic system
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"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...
== Consistancy
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.
[edit] 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?
[edit] "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. --71.246.5.61 16:22, 5 August 2006 (UTC)
