4.4. How theories develop

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.


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 :


Diverse construction schemes let new types play the roles of sets defined in set theory by operations between sets named by previous types: Each of the following schemes suffices to produce both others through the use of previous ones: The last one can be obtained by composing constructions of products (to collapse bound variables as one, and the same for parameters), a definition and a type of unary relations or functions (either as particular relations, or by adapting the form of axioms, replacing ⇔ by = ).

A development scheme at each level looks like a component at the next level

Despite their differences of nature, the 3 levels of development have remarkable similarities:

A proof of a theorem A in a theory T looks like a contradiction in the theory (T∪ ¬A); a contradiction (4th kind of component) in a theory is a proof whose theorem is 0 (which would be a very bad axiom).

Definitions look like axioms: a definition of a predicate is given by a formula with free variables; axioms are chosen from ground formulas. A ground formula would only define a Boolean constant (whose value depends on the model), which do not play any effective role of structures of the described system (giving roles to objects).
A predicate defined by a provable formula (with free variables, but that can bound by ∀ to appear as theorem), would be constantly true and thus not structuring either.

A construction of a type of structures is given by a variable structure, that is a structure symbol with a selection of variables to play the role of parameters. Structures in the language are invariant, i.e. without parameter.
A type of structures constructed by what behaves as an invariant structure (not varying with the value of parameters) would be a singleton, thus worthless.

How constructions preserve isomorphisms

The sense in which constructions preserve models, is more subtle than with definitions: any model of T is extensible to a model of the extended theory T' = TXRA (where X is a type, R is a pack of structures and A is a pack of axioms), but any identical copy of this extension of interpretation would fit as well. But the extension is «essentially unique» in the sense that any 2 possible ones will be the mere copy of each other, being related by a unique isomorphism (like initial or final objects).
Generally, for any isomorphism f between models E and F of T, with respective extensions as models E' and F' of extended theory T', we may have several cases Indeed in this case, any 2 isomorphisms g,h from E' to F' with the same restriction to E, will be equal because
The automorphism h−1g of E' extended by IdX' forms an automorphism of the model (E'E'') of T''. As f is invariant, it stays equal to its image f০(h−1g)−1 by this automorphism. Therefore h−1g = IdE' , i.e. g=h.
What we can see here strictly preserved by constructions, is the class of isomorphisms as an abstract category.
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. How theories develop
4.5. Second-order logic
4.6. Well-foundedness
4.7. Ordinals and cardinals
4.8. Undecidability of the axiom of choice
4.9. Second-order arithmetic
4.10. The Incompleteness Theorem