2
votes
1answer
31 views

Are all theorems of minimal arithmetic theorems of a given theory?

I am working on some metamathematics revision and the following question came up. Let the theory $R_0$ be axiomatized by the following axiom schemata which hold for all $n,m \in \mathbb{N}$: ...
2
votes
3answers
56 views

Satisfiability Proof Question

Exercise: Prove that $\Gamma\models A$ iff $\Gamma\cup\{\neg A\}$ is not satisfiable. Proof: We must prove two clauses: $\Gamma\models A\Rightarrow \Gamma\cup\{\neg A\}$ is not satisfiable ...
5
votes
1answer
98 views

Do metatheoretic results carry between mutually interpretable theories?

If two theories A and B are mutually interpretable, in the sense of there existing a translation procedure from A to B and from B to A, does it follow that whatever metatheoretic results (e.g., ...
0
votes
2answers
128 views

Is there a useful application of Peano arithmetic?

If there is, can someone provide an example of how Peano arithmetic can be used to solve a real-world problem? If not, can someone provide an example of any axiomatic system other than ZFC that can ...
2
votes
1answer
240 views

Mathematical structures and signature

From Wikipedia: In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach (or relate) to the set, making it easier ...
7
votes
1answer
134 views

Are there any strongly axiomatizable logics that are not compact?

I mean here a logic in the sense of a language and semantics. By strongly axiomatizable I mean strongly sound and strongly complete. So I'm basically asking if there is a particular deductive system ...