View source for Meta-theorem

A statement about a formal axiomatic theory, obtained within a definite [[Meta-theory|meta-theory]].



====Comments====
In category theory (cf. [[Category|Category]]), the term  "meta-theorem"  has acquired a more specific meaning: it refers to an assertion of the form  "if P is any statement (expressed in an appropriate formal language) about categories of a given type (e.g. Abelian categories or regular categories), then the validity of P in some particular category (e.g. the category of Abelian groups or the category of sets) implies its validity in all categories of the given type" . Results of this kind are generally deduced from imbedding theorems which assert that any category of the given type can be imbedded (in a structure-preserving way) into (a power of) the particular category under consideration.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Freyd,   "Abelian categories: An introduction to the theory of functors" , Harper &amp; Row  (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Freyd,   A. Scedrov,   "Geometric logic" , North-Holland  (1989)</TD></TR></table>