metatheories, general theories and axiomsGeneralTheoriesAndAxiomsMetatheories2009-05-29 11:36:362009-05-29 11:36:36Topicmeta-theorymetatheorymetalogicmetamathematicsmeta-theoremsmetatheoremsmetatheoriesaxioms and general theories% almost certainly you want these
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\newcommand{\hr}{{\hookrightarrow}}This is a topic on the applications to axiomatics in theoretical physics of meta-theories, metalogic and metamathematics. The topic is of potential importance for axiomatic approaches in mathematical physics in areas such as axiomatic quantum field theory (AX-QFT), local quantum field theories (AQFT), general relativity theory, general dynamics system theories and axiomatic mathematical biophysics or abstract relational biology. The approach has and had wide applications to both in metalogic, mathematics, metamathematics.
\subsection{Metatheory, Meta-Logic and Meta-Mathematics}
A {\em methatheory} or {\em meta-theory} can be described as a higher level theory about theories belonging to a lower theory class $\mathbb{M}$, or first-level theories. With this meaning, a theory $\mathcal{T}$ of the domain $\mathcal{D}$ is a meta-theory if $\mathcal{D}$ is a theory belonging to a class $\mathbb{M}$ of (lower-level, or first level) theories. A general theory is not a meta-theory because its domain $\mathcal{D}$ does not contain any other theories. Valid statements made in a meta-theory are called {\em meta-theorems} or {\em metatheorems}.
A {\em metalogic} is then a meta-theory of various types of logic.
{\em Meta-mathematics} is concerned with the study of metatheories containing mathematical metatheorems.
As an example of a meta-theory is the theory of super-categories $\mathcal{S}$ concerned with metatheorems about categories of categories. On the other hand, an example of a metatheory of supercategories $\S$, such as organismic supercategories $OS$, is the metatheory of the higher dimensional supercategory of supercategories. Higher dimensional algebra (HDA) is a metatheory of algebraic categories and other algebraic structures; good examples are double groupoids, double algebroids and their categories, as well as double categories. Further specific examples of HDA are 2-Lie groups and 2-Lie algebras, as well as their categories of 2-Lie groups and 2-Lie algebras.
In the perspective of the development of mathematics, advances in logic --and over the last century in logics and meta-logics -- have played, and are playing, very important roles both in the foundations of mathematics, as well as in related areas such as: categorical logics, many-valued logic algebras, model theory and many specific fields of mathematics including, but not limited to, number theory/arithmetics.