Categoric system of axioms
categorical system of axioms
Any system of axioms 
for which all models of the signature of 
satisfying these axioms are isomorphic. It follows from the Mal'tsev–Tarski theorem on elementary extensions that models of a categorical first-order system 
of axioms have finite cardinality. The converse also holds: For any finite model 
there exists a categorical first-order system 
of axioms whose models are isomorphic to 
. Let 
be the set of universal closures of the formulas
1) 
;
2) 
;
3) 
;
4) 
;
5) 
;
6) 
;
7) 
, where 
is any formula of signature 
.
This system of axioms 
is known under the name of Peano arithmetic. The model 
of natural numbers is a model for 
. However, there exists a model of 
that is not isomorphic to 
. Let 
be the system obtained from 
by replacing the scheme of elementary induction 7) by the axiom of complete induction
 |
written in a second-order language. Then the system 
is categorical and all models of 
are isomorphic to 
. Another method of categorical description of 
consists in appending to 
the following infinite axiom (of the language 
):
 |
when 
is short for the sum 
of 
ones.
References
| [1] | J.R. Shoenfield, "Mathematical logic" , Addison-Wesley (1967) |
Categoric system of axioms. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Categoric_system_of_axioms&oldid=43466