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''categorical system of axioms''

Any system of axioms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c0207201.png" /> for which all models of the signature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c0207202.png" /> satisfying these axioms are isomorphic. It follows from the Mal'tsev–Tarski theorem on elementary extensions that models of a categorical first-order system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c0207203.png" /> of axioms have finite cardinality. The converse also holds: For any finite model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c0207204.png" /> there exists a categorical first-order system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c0207205.png" /> of axioms whose models are isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c0207206.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c0207207.png" /> be the set of universal closures of the formulas

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c0207208.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c0207209.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072010.png" />;

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072011.png" />;

5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072012.png" />;

6) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072013.png" />;

7) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072015.png" /> is any formula of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072016.png" />.

This system of axioms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072017.png" /> is known under the name of Peano arithmetic. The model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072018.png" /> of natural numbers is a model for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072019.png" />. However, there exists a model of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072020.png" /> that is not isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072021.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072022.png" /> be the system obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072023.png" /> by replacing the scheme of elementary induction 7) by the axiom of complete induction

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072024.png" /></td> </tr></table>

written in a second-order language. Then the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072025.png" /> is categorical and all models of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072026.png" /> are isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072027.png" />. Another method of categorical description of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072028.png" /> consists in appending to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072029.png" /> the following infinite axiom (of the language <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072030.png" />):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072031.png" /></td> </tr></table>

when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072032.png" /> is short for the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020720/c02072034.png" /> ones.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.R. Shoenfield,   "Mathematical logic" , Addison-Wesley  (1967)</TD></TR></table>