Essentially-undecidable theory
An algorithmically-undecidable logical theory, all consistent extensions of which are also undecidable (see Undecidability). An elementary theory is an essentially-undecidable theory if and only if every model of it has an undecidable elementary theory. Every complete undecidable theory is an essentially-undecidable theory, as is formal arithmetic (cf. Arithmetic, formal); no theory with a finite model can be an essentially-undecidable theory.
The essential undecidability of a suitable finitely-axiomatizable elementary theory 
is often used in proving the undecidability of a given theory 
(see [1], [2]). In this proof, 
is interpreted in any model 
of 
. The domain of interpretation and the values of the elements of the signature of 
are defined using values in the model 
of suitable formulas in the language of 
. If the interpretation is a model of 
, then 
is undecidable; moreover, this theory is hereditarily undecidable, i.e. all of its subtheories of the same signature as 
are undecidable. This method is used to prove the undecidability of elementary predicate logic, elementary group theory, elementary field theory, etc. Finitely-axiomatized formal arithmetic is often used as the essentially-undecidable theory 
.
References
| [1] | A. Tarski, A. Mostowski, R.M. Robinson, "Undecidable theories" , North-Holland (1953) |
| [2] | Yu.L. Ershov, I.A. Lavrov, A.D. Taimanov, M.A. Taitslin, "Elementary theories" Russian Math. Surveys , 20 : 4 (1965) pp. 35–105 Uspekhi Mat. Nauk , 20 : 4 (1965) pp. 37–108 |
| [3] | S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951) |
Essentially-undecidable theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Essentially-undecidable_theory&oldid=42137