Does there exist a consistent theory (formal system) which has, say
- a set of axioms $\{A_1,A_2,\dots, A_n\}$ + rules of inference
such that there is
- a proper subset of (non-trivial) theorems $\{T_1,T_2, \dots,T_n\}$
which if taken themselves as axioms with the same rules would reproduce the whole theory itself?
In naive words: is there a consistent theory which contains "itself" as a part in a self-similar (like a fractal) manner?
Is ZFC such a theory?
Every single propositional calculus which has CpCqp (this is in Polish notation) as an axiom has this property (or stronger system, so for example, any first-order predicate calculus also has this property).
A consequence solely of CpCqp is CpCqCrq. But, from CpCqCrq, CpCqp can get derived in a single detachment. Thus, anytime that CpCqp gets used as an axiom, CpCqCrq could replace it as an axiom and we have the same theorems (given that we have at least three variables). Unless you have a small number of variables, there are many, if not infinite also, theorems that could work similarly, since there's a sort of pattern of consequences from CpCqp which work out similarly...
CaCpCqp
CaCbCpCqp
.
.
.
Ca$_1$Ca$_2$...Ca$_n$CpCqp.
So, the answer to your question, ends up 'yes', at least without the 'non-trivial' part, of which I'm not sure what that means.