# Meta-theorem

A statement about a formal axiomatic theory, obtained within a definite meta-theory.

#### Comments

In category theory (cf. 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

[a1] | P. Freyd, "Abelian categories: An introduction to the theory of functors" , Harper & Row (1964) |

[a2] | P. Freyd, A. Scedrov, "Geometric logic" , North-Holland (1989) |

**How to Cite This Entry:**

Meta-theorem. A.G. Dragalin (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Meta-theorem&oldid=17659