4.4. Development of theories : definitions

Given a theory T (with types, structures and axioms), a development of T is a extended theory T' = T∪(pack of extra components) which remains «equivalent» to T.  This concept has 3 aspects:
There are 3 levels of development with their respective schemes:

The use of a theorem as an available axiom for the proofs of the next theorems, abbreviates a repetition of its proof as a part of the next proofs (with possible substitutions of variables by terms to which the theorem is applied).
When inserting theories in the set theoretical framework, constructions become set theoretical definitions, which is why the distinction between definitions and constructions is not usually clarified in the literature.

The Galois connection (Mod,Tru)

For any language L, the truth relation between ground formulas and systems defines a Galois connection (Mod,Tru), where Mod gives the class of models of any set A of ground formulas taken as axioms, while Tru gives the set of true formulas which all given systems have in common. Replacing the class of all systems by the set of all L-structures on ℕ, leaves unchanged and thus gives definite sense to the closure Tru০Mod. All theorems of a theory belong to the closure Tru০Mod(A) of axioms, which is the condition for a formula to leave unchanged Mod A when it is added to A.


We have 3 schemes of definitions:

How definitions preserve models

For any theory T with a model E, extensions T' of T by an additional axiom and/or structure symbol may have different effects: The uniqueness of the extension for every E, means that in the doubly extended theory (T∪{R,A,R1,A1}) where (R1,A1) is a copy of (R,A), one can prove

∀(variables), RR1

Thanks to the completeness theorem, if every model of T can be extended as a model of a T' = (T∪{R,A}) where R is a new symbol and A is a new (set of) axiom(s), then every ground formula of T provable in T', is also provable in T.
The converse may be wrong: a theory T may be extensible to a theory T' unable to prove more of its formulas, but to which not all its models are extensible. Here are some examples from non-standard models of arithmetic : Still, like proofs strictly preserve models, definitions can be seen as strictly preserving something else. For any isomorphism f between models E and F of T, with respective extensions as models E' and F' of a theory T' extending T by further structures and axioms, f may stay or not as an isomorphism between E' and F', i.e. preserve the interpretation of the new structure(s). It will be an isomorphism whenever the additional axiom(s) form a definition, determining the new structure(s) and thus making it invariant by isomorphisms.
This switch from models to isomorphisms provides similarities between levels of development, not only from proofs to definitions but also from definitions to constructions, making sense to how these also essentially preserve theories, as we shall now explain.
Set theory and foundations of mathematics
1. First foundations of mathematics
2. Set theory (continued)
3. Algebra 1
4. Model Theory
4.1. Finiteness and countability
4.2. The Completeness Theorem
4.3. Non-standard models of Arithmetic
4.4. Development of theories : definitions
4.5. Constructions
4.6. Second-order logic
4.7. Well-foundedness
4.8. Ordinals and cardinals
4.9. Undecidability of the axiom of choice
4.10. Second-order arithmetic
4.11. The Incompleteness Theorem